Απειροστικός Λογισμός Ι - Σημειώσεις

Apeirostikìc Logismìc I

Prìqeirec Shmei¸seic

Tm ma Majhmatik¸nPanepist mio Ajhn¸n

Aj na, 2006

Perieqìmena

1 To sÔnolo twn pragmatik¸n arijm¸n 11.1 FusikoÐ arijmoÐ 1

1.1aþ Arq tou elaqÐstou kai arq thc epagwg c 11.1bþ Diairetìthta 4

1.2 RhtoÐ arijmoÐ 51.2aþ S¸mata 51.2bþ Diatetagmèna s¸mata 71.2gþ RhtoÐ arijmoÐ 8

1.3 PragmatikoÐ arijmoÐ 91.3aþ H arq thc plhrìthtac 91.3bþ Qarakthrismìc tou supremum 13

1.4 Sunèpeiec tou axi¸matoc thc plhrìthtac 131.4aþ 'Uparxh n�ost c rÐzac 131.4bþ Arqim deia idiìthta 171.4gþ 'Uparxh akeraÐou mèrouc 171.4dþ Puknìthta twn rht¸n kai twn arr twn stouc pragmati-

koÔc arijmoÔc 181.5 OrismoÐ kai sumbolismìc 19

1.5aþ Apìluth tim 191.5bþ To sÔnolo twn epektetamènwn pragmatik¸n arijm¸n 201.5gþ Diast mata 21

1.6 Anisìthtec 221.7 *Par�rthma: Tomèc Dedekind 241.8 Ask seic 26

2 AkoloujÐec pragmatik¸n arijm¸n 332.1 AkoloujÐec pragmatik¸n arijm¸n 332.2 SÔgklish akolouji¸n 34

2.2aþ Orismìc tou orÐou 342.2bþ AkoloujÐec pou teÐnoun sto �peiro 372.2gþ H �rnhsh tou orismoÔ 38

2.3 'Algebra twn orÐwn 382.4 Basik� ìria kai basik� krit ria sÔgklishc 42

2.4aþ Basik� ìria 42

iv · Perieqomena

2.4bþ Krit rio thc rÐzac kai krit rio tou lìgou 432.5 SÔgklish monìtonwn akolouji¸n 45

2.5aþ SÔgklish monìtonwn akolouji¸n 452.5bþ O arijmìc e 462.5gþ Arq twn kibwtismènwn diasthm�twn 482.5dþ Anadromikèc akoloujÐec 49

2.6 Je¸rhma Bolzano-Weierstrass 502.6aþ UpakoloujÐec 502.6bþ Je¸rhma Bolzano-Weierstrass 512.6gþ DeÔterh apìdeixh me qr sh thc arq c tou kibwtismoÔ 52

2.7 AkoloujÐec Cauchy 532.8 *An¸tero kai kat¸tero ìrio akoloujÐac 562.9 *Par�rthma: suz thsh gia to axÐwma thc plhrìthtac 592.10 Ask seic 61

3 SuneqeÐc sunart seic 693.1 Sunart seic 69

3.1aþ Ask seic 723.2 Sunèqeia sun�rthshc 75

3.2aþ Orismìc thc sunèqeiac 753.2bþ H �rnhsh tou orismoÔ 763.2gþ Arq thc metafor�c 773.2dþ Sunèqeia kai pr�xeic metaxÔ sunart sewn 783.2eþ Sunèqeia kai topik sumperifor� 79

3.3 Basik� jewr mata gia suneqeÐc sunart seic 803.3aþ To je¸rhma el�qisthc kai mègisthc tim c 803.3bþ To je¸rhma endi�meshc tim c 823.3gþ ParadeÐgmata 853.3dþ Efarmogèc twn basik¸n jewrhm�twn 863.3eþ Sunèqeia antÐstrofhc sun�rthshc 87

4 'Oria sunart sewn 914.1 ShmeÐa suss¸reushc 914.2 Orismìc tou orÐou 92

4.2aþ Arq thc metafor�c 954.3 Sqèsh orÐou kai sunèqeiac 964.4 Trigwnometrikèc sunart seic 984.5 Ask seic gia ta Kef�laia 3 kai 4 100

5 Par�gwgoc 1075.1 Orismìc thc parag¸gou 1075.2 Kanìnec parag¸gishc 109

5.2aþ Kanìnac thc alusÐdac 1105.2bþ Par�gwgoc antÐstrofhc sun�rthshc. 1115.2gþ Par�gwgoi an¸terhc t�xhc 112

5.3 Krisima shmeia 112

Perieqomena · v

5.4 Jewrhma meshc timhc 1155.5 Aprosdioristec morfec 1195.6 Idiothta Darboux 1225.7 Gewmetrikh shmasia thc deuterhc paragwgou 123

5.7aþ Kurtec kai koilec sunarthseic 1235.7bþ Asumptwtec 126

5.8 Ask seic 127

6 UpodeÐxeic gia tic Ask seic 1316.1 PragmatikoÐ ArijmoÐ 1316.2 AkoloujÐec pragmatik¸n arijm¸n 1436.3 AkoloujÐec pragmatik¸n arijm¸n 1596.4 Sunart seic 1756.5 Sunèqeia kai 'Oria 1796.6 Par�gwgoc kai melèth sunart sewn 192

Kef�laio 1

To sÔnolo twnpragmatik¸n arijm¸n

1.1 FusikoÐ arijmoÐ

H austhr jemelÐwsh tou sunìlou N = {1, 2, 3, . . .} twn fusik¸n arijm¸ngÐnetai mèsw twn axiwm�twn tou Peano. 'Eqontac dedomèno to N, mporoÔmena d¸soume austhr kataskeu tou sunìlou Z twn akeraÐwn arijm¸n kai tousunìlou Q twn rht¸n arijm¸n. JewroÔme ìti o anagn¸sthc eÐnai exoikeiwmènocme tic pr�xeic kai th di�taxh sta sÔnola twn fusik¸n, twn akeraÐwn kai twnrht¸n arijm¸n. Se aut th sÔntomh par�grafo suzht�me k�poiec basikèc arqècgia touc fusikoÔc arijmoÔc.

1.1aþ Arq tou elaqÐstou kai arq thc epagwg c

Arq tou elaqÐstou. K�je mh kenì sÔnolo S fusik¸n arijm¸n èqei el�qi-sto stoiqeÐo. Dhlad , up�rqei a ∈ S me thn idiìthta: a ≤ b gia k�je b ∈ S.

H arq tou elaqÐstou èqei wc sunèpeia thn ex c prìtash:

Je¸rhma 1.1.1. Den mporoÔme na epilèxoume �peirouc to pl joc fusikoÔcarijmoÔc oi opoÐoi na fjÐnoun gnhsÐwc.

Apìdeixh. Ac upojèsoume ìti up�rqei mia tètoia epilog fusik¸n arijm¸n:

n1 > n2 > · · · > nk > nk+1 > · · · .

Apì thn arq tou elaqÐstou, to sÔnolo S = {nk : k ∈ N} èqei el�qisto stoiqeÐo:autì ja eÐnai thc morf c nm gia k�poion m ∈ N. 'Omwc, nm+1 < nm kainm+1 ∈ S, to opoÐo eÐnai �topo. 2

Mia deÔterh sunèpeia thc arq c tou elaqÐstou eÐnai h arq thc epagwg c:

Je¸rhma 1.1.2 (arq thc epagwg c). 'Estw S èna sÔnolo fusik¸narijm¸n me tic ex c idiìthtec:

2 · To sunolo twn pragmatikwn arijmwn

(i) O 1 an kei sto S.

(ii) An k ∈ S tìte k + 1 ∈ S.

Tìte, to S tautÐzetai me to sÔnolo ìlwn twn fusik¸n arijm¸n: S = N.

Apìdeixh. Jètoume T = N \ S (to sumpl rwma tou S) kai upojètoume ìti to T

eÐnai mh kenì. Apì thn arq tou elaqÐstou, to T èqei el�qisto stoiqeÐo to opoÐosumbolÐzoume me a. AfoÔ 1 ∈ S, anagkastik� èqoume a > 1 opìte a − 1 ∈ N.AfoÔ o a tan to el�qisto stoiqeÐo tou T , èqoume a− 1 ∈ S. Apì thn upìjesh(ii),

a = (a− 1) + 1 ∈ S.

Katal xame se �topo, �ra to T eÐnai to kenì sÔnolo. Sunep¸c, S = N. 2

Parat rhsh. H arq tou elaqÐstou kai to Je¸rhma 1.1.2 eÐnai logik� isodÔnamecprot�seic. Prospaj ste na apodeÐxete thn isodunamÐa touc.

H arq thc peperasmènhc epagwg c mac epitrèpei na apodeiknÔoume ìti k�poiaprìtash P (n) pou afor� touc fusikoÔc arijmoÔc isqÔei gia k�je n ∈ N. ArkeÐ naelègxoume ìti h P (1) isqÔei (aut eÐnai h b�sh thc epagwg c) kai na apodeÐxoumeth sunepagwg P (k) ⇒ P (k + 1) (autì eÐnai to epagwgikì b ma). ParadeÐgmataprot�sewn pou apodeiknÔontai me th {mèjodo thc majhmatik c epagwg c} jasunant�me se ìlh th di�rkeia tou maj matoc.

Je¸rhma 1.1.3 (mèjodoc thc epagwg c). 'Estw Π(n) mia (majhmati-k ) prìtash pou exart�tai apì ton fusikì n. An h Π(1) alhjeÔei kai gia k�jek ∈ N èqoume

Π(k) alhj c =⇒ Π(k + 1) alhj c ,

tìte h Π(n) alhjeÔei gia k�je fusikì n.

Apìdeixh. To sÔnolo S = {n ∈ N : Π(n) alhj c} ikanopoieÐ tic upojèseic touJewr matoc 1.1.2. 'Ara, S = N. Autì shmaÐnei ìti h Π(n) alhjeÔei gia k�jefusikì n. 2

AxÐzei na anafèroume dÔo parallagèc tou Jewr matoc 1.1.2. H apìdeixh toucaf netai san �skhsh gia ton anagn¸sth (mimhjeÐte thn prohgoÔmenh apìdeixh -qrhsimopoi ste thn arq tou elaqÐstou).

Je¸rhma 1.1.4. 'Estw m ∈ N kai èstw S èna sÔnolo fusik¸n arijm¸n metic ex c idiìthtec: m ∈ S kai an k ∈ S me k ≥ m tìte k + 1 ∈ S. Tìte,S ⊇ {n ∈ N : n ≥ m} = {m,m + 1, . . .}. 2

Je¸rhma 1.1.5. 'Estw S èna sÔnolo fusik¸n arijm¸n me tic ex c idiìthtec:1 ∈ S kai an 1, . . . , k ∈ S tìte k + 1 ∈ S. Tìte, S = N. 2

IsodÔnama, èqoume ta ex c:

Je¸rhma 1.1.6. 'Estw Π(n) mia prìtash pou exart�tai apì ton fusikì n. Anh Π(m) alhjeÔei gia k�poion m ∈ N kai an gia k�je k ≥ m isqÔei h sunepagwg

Π(k) alhjeÔei =⇒ Π(k + 1) alhjeÔei,

tìte h Π(n) alhjeÔei gia k�je fusikì n ≥ m.

1.1 Fusikoi arijmoi · 3

Je¸rhma 1.1.7. 'Estw Π(n) mia prìtash pou exart�tai apì to fusikì n. Anh Π(1) alhjeÔei kai an gia k�je k ∈ N isqÔei h sunepagwg

oi Π(1), . . . , Π(k) alhjeÔoun =⇒ Π(k + 1) alhjeÔei,

tìte h Π(n) alhjeÔei gia k�je fusikì n.

H ènnoia thc {majhmatik c prìtashc} eÐnai bebaÐwc amfilegìmenh. Gia par�-deigma, sqoli�ste to ex c par�deigma apì to biblÐo {Eisagwg sthn 'Algebra}tou J. B. Fraleigh: jèloume na deÐxoume ìti k�je fusikìc arijmìc èqei k�poia en-diafèrousa idiìthta (h opoÐa ton xeqwrÐzei apì ìlouc touc upìloipouc). JètoumeΠ(n) thn prìtash {o n èqei k�poia endiafèrousa idiìthta} kai qrhsimopoioÔmeth mèjodo thc epagwg c:

H Π(1) alhjeÔei diìti o 1 eÐnai o monadikìc fusikìc arijmìc pou isoÔtaime to tetr�gwnì tou (aut h idiìthta eÐnai endiafèrousa, afoÔ xeqwrÐzei ton 1apì ìlouc touc �llouc fusikoÔc arijmoÔc). Upojètoume ìti oi Π(1), . . . , Π(k)alhjeÔoun. An h Π(k + 1) den tan alhj c, tìte o k + 1 ja tan o mikrìterocfusikìc arijmìc pou den èqei kamÐa endiafèrousa idiìthta, k�ti pou eÐnai apìmìno tou polÔ endiafèron. 'Ara, h Π(k + 1) alhjeÔei. SÔmfwna me to Je¸rhma1.1.7, h Π(n) alhjeÔei gia k�je n ∈ N.

Paradeigmata.

(a) Exet�ste gia poiec timèc tou fusikoÔ arijmoÔ n isqÔei h anisìthta 2n > n3.

Sqìlio. An k�nete arketèc dokimèc ja peisteÐte ìti h 2n > n3 isqÔei gia n = 1,den isqÔei gia n = 2, 3, . . . , 9 kai (m�llon) isqÔei gia k�je n ≥ 10.

DeÐxte me epagwg (qrhsimopoi¸ntac to Je¸rhma 1.1.6) ìti h 2n > n3 isqÔeigia k�je n ≥ 10: gia to epagwgikì b ma upojètoume ìti h 2m > m3 isqÔei giak�poion m ≥ 10. Tìte,

2m+1 > 2m3 > (m + 1)3

an isqÔei h anisìthta1 + 3m + 3m2 < m3.

'Omwc, afoÔ m ≥ 10, èqoume

1 + 3m + 3m2 < m2 + 3m2 + 3m2 = 7m2 < m ·m2 = m3.

(b) Na deiqjoÔn me epagwg oi tautìthtec

1 + 2 + · · ·+ n =n(n + 1)

2,

12 + 22 + · · ·+ n2 =n(n + 1)(2n + 1)

6,

1 + 3 + · · ·+ (2n− 1) = n2.

Sqìlio. H apìdeixh (me th mèjodo thc epagwg c) den parousi�zei kamÐa duskolÐaapì th stigm pou mac dÐnetai h ap�nthsh (to dexiì mèloc). Prospaj ste na breÐ-te mia {mèjodo} me thn opoÐa na gr�fete se {kleist morf } ìla ta ajroÐsmatathc morf c

S(n, k) = 1k + 2k + · · ·+ nk.

(g) DeÐxte ìti èna sÔnolo S me n stoiqeÐa èqei akrib¸c 2n uposÔnola.Apìdeixh. Jèloume na deÐxoume me epagwg thn prìtash

Π(n): An to S èqei n stoiqeÐa tìte to S èqei akrib¸c 2n uposÔnola.

An n = 1 tìte to S eÐnai monosÔnolo kai èqei akrib¸c dÔo uposÔnola, to ∅ kaito S. Sunep¸c, h Π(1) alhjeÔei.

Upojètoume ìti h Π(k) alhjeÔei. 'Estw S = {x1, . . . , xk, xk+1} èna sÔnolome (k + 1) stoiqeÐa. JewroÔme to sÔnolo

T = S \ {xk+1} = {x1, . . . , xk}.To T èqei k stoiqeÐa, opìte èqei 2k uposÔnola. T¸ra, k�je uposÔnolo tou S

ja perièqei den ja perièqei to xk+1. Ta uposÔnola tou S pou den perièqoun toxk+1 eÐnai akrib¸c ta uposÔnola tou T , dhlad to pl joc touc eÐnai 2k. Apì thn�llh pleur�, k�je uposÔnolo tou S pou perièqei to xk+1 prokÔptei apì k�poiouposÔnolo tou T me thn prosj kh tou xk+1 (antÐstrofa, k�je uposÔnolo touT prokÔptei apì k�poio uposÔnolo tou S pou perièqei to xk+1 me thn afaÐreshtou xk+1). Dhlad , to pl joc twn uposunìlwn tou S pou perièqoun to xk+1

eÐnai 2k (ìsa eÐnai ta uposÔnola tou T ). 'Epetai ìti to sunolikì pl joc twnuposunìlwn tou S eÐnai

2k + 2k = 2 · 2k = 2k+1.

Dhlad , h Π(k + 1) alhjeÔei.Sunep¸c, h Π(n) alhjeÔei gia k�je n ∈ N. 2

1.1bþ Diairetìthta

'Estw a, b ∈ Z. Lème ìti o a diaireÐ ton b kai gr�foume a | b, an up�rqeix ∈ Z ¸ste b = ax. Se aut thn perÐptwsh ja lème ìti o a eÐnai diairèthc toub ìti o b eÐnai pollapl�sio tou a.

Je¸rhma 1.1.8 (tautìthta thc diaÐreshc). Upojètoume ìti a ∈ N kaib ∈ Z. Tìte, up�rqoun monadikoÐ q, r ∈ Z ¸ste

b = aq + r kai 0 ≤ r < a.

{Gewmetrik apìdeixh}: 'Enac aplìc gewmetrikìc trìpoc gia na skeftìmastethn tautìthta thc diaÐreshc eÐnai o ex c: fantazìmaste mia eujeÐa p�nw sthnopoÐa èqoume shmei¸sei me koukÐdec touc akeraÐouc. Shmei¸noume me pio skoÔreckoukÐdec ta pollapl�sia tou a. Diadoqikèc skoÔrec koukÐdec èqoun apìstashakrib¸c Ðsh me a. Tìte, èna apì ta dÔo sumbaÐnei:

(i) O akèraioc b pèftei p�nw se k�poia apì autèc tic skoÔrec koukÐdec, opìteo b eÐnai pollapl�sio tou a kai r = 0.

(ii) O akèraioc b brÐsketai an�mesa se dÔo diadoqikèc skoÔrec koukÐdec, dhlad an�mesa se dÔo diadoqik� pollapl�sia tou a, kai h apìstash r an�mesaston b kai to megalÔtero pollapl�sio tou a pou eÐnai mikrìtero apì ton b

eÐnai ènac jetikìc akèraioc pou den xepern�ei ton a− 1.

1.2 Rhtoi arijmoi · 5

H austhr apìdeixh pou ja d¸soume parak�tw basÐzetai se aut thn idèa: je-wroÔme to sÔnolo S twn {apost�sewn} b − as tou b apì tic skoÔrec koukÐdecpou brÐskontai arister� tou. ExasfalÐzoume ìti eÐnai mh kenì, �ra èqei el�qistostoiqeÐo b − aq. H koukÐda aq eÐnai aut pou brÐsketai amèswc prin apì ton b,kai h apìstash r = b− aq prèpei na eÐnai mikrìterh apì a.

Apìdeixh tou Jewr matoc 1.1.8. ApodeiknÔoume pr¸ta thn Ôparxh arijm¸n q, r ∈Z pou ikanopoioÔn to zhtoÔmeno. JewroÔme to sÔnolo

S = {b− as : s ∈ Z} ∩ Z+

twn mh arnhtik¸n akeraÐwn thc morf c b−as. Den eÐnai dÔskolo na doÔme ìti toS eÐnai mh kenì: an b ≥ 0, tìte b−a·0 ∈ S. An b < 0, tìte b−ab = (1−a)b ∈ Z+.

Apì thn arq tou elaqÐstou to S èqei el�qisto stoiqeÐo, to opoÐo sumbo-lÐzoume me r. Apì ton orismì tou S èqoume r ≥ 0 kai up�rqei q ∈ Z ¸steb− aq = r. Mènei na deÐxoume ìti r < a. Ac upojèsoume ìti r ≥ a. Tìte,

b− a(q + 1) = b− aq − a = r − a ≥ 0,

dhlad , b − a(q + 1) ∈ S. 'Omwc b − a(q + 1) = r − a < r, to opoÐo eÐnai �topoafoÔ o r tan to el�qisto stoiqeÐo tou S.

ApodeiknÔoume t¸ra th monadikìthta twn q kai r. Ac upojèsoume ìti

b = aq1 + r1 = aq2 + r2,

ìpou 0 ≤ r1, r2 < a. QwrÐc periorismì thc genikìthtac upojètoume ìti r1 ≥ r2

(opìte q1 ≤ q2). Tìte,r1 − r2 = a(q2 − q1).

An q1 < q2, tìte a(q2 − q1) ≥ a en¸ r1 − r2 < a. 'Eqoume antÐfash, �ra q1 = q2

kai r1 = r2. 2

ShmeÐwsh. Apì to Je¸rhma 1.1.8, k�je akèraioc b gr�fetai monos manta sthmorf b = 2q + r gia k�poion q ∈ Z kai k�poion r ∈ {0, 1}. Lème ìti o b eÐnai�rtioc an r = 0. An r = 1, tìte lème ìti o b eÐnai perittìc. Parathr ste ìtiopoiad pote dÔnamh perittoÔ akeraÐou eÐnai perittìc akèraioc.

1.2 RhtoÐ arijmoÐ

1.2aþ S¸mata

JewroÔme èna mh kenì sÔnolo Σ efodiasmèno me dÔo pr�xeic + (thn prìsje-sh) kai · (ton pollaplasiasmì) oi opoÐec ikanopoioÔn ta ex c:

(a) Axi¸mata thc prìsjeshc. Gia k�je zeug�ri x, y stoiqeÐwn tou Σup�rqei akrib¸c èna stoiqeÐo tou Σ pou sumbolÐzetai me x + y kai lègetai �-jroisma twn x, y. H pr�xh pou stèlnei to zeug�ri (x, y) sto x + y lègetaiprìsjesh kai èqei tic akìloujec idiìthtec:

• Prosetairistikìthta: gia k�je x, y, z ∈ Σ isqÔei (x + y) + z = x + (y + z).


• Antimetajetikìthta: gia k�je x, y ∈ Σ isqÔei x + y = y + x.

• Up�rqei èna stoiqeÐo tou Σ pou sumbolÐzetai me 0, ¸ste, gia k�je x ∈ Σ,

x + 0 = 0 + x = x.

• Gia k�je x ∈ Σ up�rqei èna stoiqeÐo tou Σ pou sumbolÐzetai me −x, ¸ste

x + (−x) = (−x) + x = 0.

Lème ìti to Σ me thn pr�xh thc prìsjeshc eÐnai antimetajetik om�da. DeÐxteìti to 0 kai to −x (dojèntoc tou x) orÐzontai monos manta. O −x eÐnai oantÐjetoc tou x. H afaÐresh sto Σ orÐzetai apì thn

x− y = x + (−y) (x, y,∈ Σ).

(b) Axi¸mata tou pollaplasiasmoÔ. Gia k�je zeug�ri x, y ∈ Σ up�rqeiakrib¸c èna stoiqeÐo tou Σ pou sumbolÐzetai me x · y (gia aplìthta, xy) kailègetai ginìmeno twn x, y. H pr�xh pou stèlnei to zeug�ri (x, y) sto xy

lègetai pollaplasiasmìc kai èqei tic akìloujec idiìthtec:

• Prosetairistikìthta: gia k�je x, y, z ∈ Σ isqÔei (xy)z = x(yz).

• Antimetajetikìthta: gia k�je x, y ∈ Σ isqÔei xy = yx.

• Up�rqei èna stoiqeÐo tou Σ, diaforetikì apì to 0, pou sumbolÐzetai me 1,¸ste, gia k�je x ∈ Σ,

x1 = 1x = x.

• Gia k�je x ∈ Σ me x 6= 0 up�rqei èna stoiqeÐo tou Σ pou sumbolÐzetai mex−1, ¸ste

xx−1 = x−1x = 1.

DeÐxte ìti to 1 kai to x−1 (dojèntoc tou x 6= 0) orÐzontai monos manta. O x−1

eÐnai o antÐstrofoc tou x 6= 0. H diaÐresh sto Σ orÐzetai apì thn

x

y= xy−1 (x, y ∈ Σ, y 6= 0).

(g) H epimeristik idiìthta sundèei ton pollaplasiasmì me thn prìsjesh:gia k�je x, y, z ∈ Σ, èqoume

x(y + z) = xy + xz.

Orismìc 1.2.1. Mia tri�da (Σ,+, ·) pou ikanopoieÐ ta parap�nw lègetai s¸-ma.

1.2 Rhtoi arijmoi · 7

Parat rhsh 1.2.2. Parathr ste ìti an mia tri�da (Σ, +, ·) eÐnai s¸ma, tìteto sÔnolo Σ èqei toul�qiston dÔo stoiqeÐa: to {oudètero stoiqeÐo} 0 thc prì-sjeshc kai to {oudètero stoiqeÐo} 1 tou pollaplasiasmoÔ. MporoÔme m�listana d¸soume par�deigma s¸matoc (Σ,+, ·) sto opoÐo aut� na eÐnai ta mìna stoi-qeÐa tou Σ: jètoume Σ = {0,1} kai orÐzoume prìsjesh kai pollaplasiasmì stoΣ jètontac

0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 0

kai0 · 0 = 0, 0 · 1 = 0, 1 · 0 = 0, 1 · 1 = 1.

Elègxte ìti me autèc tic pr�xeic to {0,1} ikanopoieÐ ta axi¸mata (a)�(g) tous¸matoc.

1.2bþ Diatetagmèna s¸mata

'Ena s¸ma (Σ, +, ·) lègetai diatetagmèno an up�rqei èna uposÔnolo Θ tou Σ,pou lègetai to sÔnolo twn jetik¸n stoiqeÐwn tou Σ, ¸ste:

• Gia k�je x ∈ Σ isqÔei akrib¸c èna apì ta akìlouja:

x ∈ Θ, x = 0, −x ∈ Θ.

• An x, y ∈ Θ tìte x + y ∈ Θ kai xy ∈ Θ.

To sÔnolo Θ orÐzei mia di�taxh sto s¸ma Σ wc ex c: lème ìti x < y (isodÔnama,y > x) an kai mìno an y−x ∈ Θ. Gr�fontac x ≤ y (isodÔnama, y ≥ x) ennooÔme:eÐte x < y x = y. Apì ton orismì,

x ∈ Θ an kai mìno an x > 0.

Apì tic idiìthtec tou Θ èpontai oi ex c idiìthtec thc di�taxhc <:

• Gia k�je x, y ∈ Σ isqÔei akrib¸c èna apì ta akìlouja:

x < y, x = y, x > y.

• An x < y kai y < z, tìte x < z.

• An x < y tìte gia k�je z isqÔei x + z < y + z.

• An x < y kai z > 0, tìte xz < yz.

• 1 > 0.

H apìdeixh aut¸n twn isqurism¸n af netai san 'Askhsh gia ton anagn¸sth.

1.2gþ RhtoÐ arijmoÐ

To sÔnolo Q twn rht¸n arijm¸n eÐnai to

Q ={m

n: m ∈ Z, n ∈ N

}.

JumhjeÐte ìtim

n=

m′

n′an kai mìno an mn′ = nm′,

kai ìti oi pr�xeic + kai · orÐzontai wc ex c:

m

n+

m1

n1=

mn1 + m1n

nn1,

m

n· m1

n1=

mm1

nn1.

Tèloc,m

n<

m1

n1an kai mìno an m1n−mn1 ∈ N.

H tetr�da (Q,+, ·, <) eÐnai tupikì par�deigma diatetagmènou s¸matoc. Ta sÔ-nola N kai Z twn fusik¸n kai twn akeraÐwn (me tic gnwstèc pr�xeic) den ikano-poioÔn ìla ta axi¸mata tou s¸matoc (exhg ste giatÐ).

L mma 1.2.3. K�je rhtìc arijmìc q gr�fetai se {an�gwgh morf } q = mn ,

ìpou o monadikìc fusikìc pou diaireÐ tìso ton m ìso kai ton n eÐnai o 1.

Apìdeixh. JewroÔme to sÔnolo

E(q) ={

n ∈ N : up�rqei m ∈ Z ¸ste q =m

n

}.

To E(q) eÐnai mh kenì uposÔnolo tou N (giatÐ q ∈ Q), �ra èqei el�qisto stoiqeÐo,ac to poÔme n0. Apì ton orismì tou E(q) up�rqei m0 ∈ Z ¸ste q = m0

n0.

Ac upojèsoume ìti up�rqei fusikìc d > 1 ¸ste d | m0 kai d | n0. Tìte,up�rqoun m1 ∈ Z kai n1 ∈ N ¸ste m0 = dm1 kai n0 = dn1 > n1. Tìte,

q =m0

n0=

dm1

dn1=

m1

n1,

dhlad n1 ∈ E(q). Autì eÐnai �topo, diìti n1 < n0. 2

Anapar�stash twn rht¸n arijm¸n sthn eujeÐa. H idèa ìti oi arij-moÐ mporoÔn na jewrhjoÔn san {apost�seic} odhgeÐ se mia fusiologik antistoÐ-qish touc me ta shmeÐa miac eujeÐac. JewroÔme tuqoÔsa eujeÐa kai epilègoumeaujaÐreta èna shmeÐo thc, to opoÐo onom�zoume 0, kai èna deÔtero shmeÐo de-xi� tou 0, to opoÐo onom�zoume 1. To shmeÐo 0 paÐzei to rìlo thc arq c thc{mètrhshc apost�sewn} en¸ h apìstash tou shmeÐou 1 apì to shmeÐo 0 prosdio-rÐzei th {mon�da mètrhshc apost�sewn}. Oi akèraioi arijmoÐ mporoÔn t¸ra natopojethjoÔn p�nw sthn eujeÐa kat� profan trìpo.

MporoÔme epÐshc na topojet soume sthn eujeÐa ìlouc touc rhtoÔc arijmoÔc.Ac jewr soume, qwrÐc periorismì thc genikìthtac, ènan jetikì rhtì arijmì q.Autìc gr�fetai sth morf q = m

n , ìpou m, n ∈ N. An topojet soume ton 1n

sthn eujeÐa tìte mporoÔme na k�noume to Ðdio kai gia ton q. Autì gÐnetai wc

1.3 Pragmatikoi arijmoi · 9

ex c: jewroÔme deÔterh eujeÐa pou pern�ei apì to 0 kai p�nw thc paÐrnoume n Ðsadiadoqik� eujÔgramma tm mata me �kra 1′, . . . , n′, xekin¸ntac apì to 0. JewroÔmethn eujeÐa pou en¸nei to n′ me to 1 thc pr¸thc eujeÐoac kai fèrnoume par�llhlhproc aut n apì to shmeÐo 1′. Aut tèmnei to eujÔgrammo tm ma 01 thc pr¸thceujeÐac sto shmeÐo 1

n (kanìnac twn analogi¸n gia ìmoia trÐgwna).EÐdame loipìn ìti k�je rhtìc arijmìc antistoiqeÐ se k�poio shmeÐo thc eu-

jeÐac. To diatetagmèno s¸ma Q ja tan èna eparkèc sÔsthma arijm¸n an, antÐ-strofa, k�je shmeÐo thc eujeÐac antistoiqoÔse se k�poion rhtì arijmì. Autììmwc den isqÔei. Apì to Pujagìreio Je¸rhma, h upoteÐnousa enìc orjogwnÐoutrig¸nou me k�jetec pleurèc m kouc 1 èqei m koc x pou ikanopoieÐ thn

x2 = 12 + 12 = 2.

An k�je m koc mporoÔse na metrhjeÐ me rhtì arijmì, tìte to m koc x ja èprepena antistoiqeÐ se k�poion rhtì q.

Je¸rhma 1.2.4. Den up�rqei q ∈ Q ¸ste q2 = 2.

Apìdeixh. Upojètoume ìti up�rqei q ∈ Q ¸ste q2 = 2. Antikajist¸ntac, anqreiasteÐ, ton q me ton −q, mporoÔme na upojèsoume ìti q > 0. Tìte, o q

gr�fetai sth morf q = m/n, ìpou m,n ∈ N kai o monadikìc fusikìc arijmìcpou eÐnai koinìc diairèthc twn m kai n eÐnai o 1. Apì thn q2 = 2 sumperaÐnoumeìti m2 = 2n2, �ra o m eÐnai �rtioc (to tetr�gwno perittoÔ eÐnai perittìc). AutìshmaÐnei ìti m = 2k gia k�poion k ∈ N. Tìte n2 = 2k2, �ra o n eÐnai ki autìc�rtioc. Autì eÐnai �topo: o 2 eÐnai koinìc diairèthc twn m kai n. 2

Up�rqoun loipìn {m kh} pou den metrioÔntai me rhtoÔc arijmoÔc. An jèloumeèna sÔsthma arijm¸n to opoÐo na eparkeÐ gia th mètrhsh opoiasd pote apìstashcp�nw sthn eujeÐa, tìte prèpei na {epekteÐnoume} to sÔnolo twn rht¸n arijm¸n.

1.3 PragmatikoÐ arijmoÐ

1.3aþ H arq thc plhrìthtac

Apì th stigm pou se èna diatetagmèno s¸ma Σ èqoume orismènh th di�taxh<, mporoÔme na mil�me gia uposÔnola tou Σ pou eÐnai �nw k�tw fragmèna.

Orismìc 1.3.1. 'Estw Σ èna diatetagmèno s¸ma. 'Ena mh kenì uposÔnolo A

tou Σ lègetai

• �nw fragmèno, an up�rqei α ∈ Σ me thn idiìthta: x ≤ α gia k�jex ∈ A.

• k�tw fragmèno, an up�rqei α ∈ Σ me thn idiìthta: x ≥ α gia k�jex ∈ A.

• fragmèno, an eÐnai �nw kai k�tw fragmèno.

K�je α ∈ Σ pou ikanopoieÐ ton parap�nw orismì lègetai �nw fr�gma (antÐstoiqa,k�tw fr�gma) tou A.

Parat rhsh 1.3.2. 'Estw ∅ 6= A ⊆ Σ kai èstw α èna �nw fr�gma tou A,dhlad x ≤ α gia k�je x ∈ A. K�je stoiqeÐo α1 tou Σ pou eÐnai megalÔtero Ðso tou α eÐnai epÐshc �nw fr�gma tou A: an x ∈ A tìte x ≤ α ≤ α1. TeleÐwcan�loga, an ∅ 6= A ⊆ Σ kai an α eÐnai èna k�tw fr�gma tou A, tìte k�je stoiqeÐoα1 tou Σ pou eÐnai mikrìtero Ðso tou α eÐnai epÐshc k�tw fr�gma tou A.

Orismìc 1.3.3. (a) 'Estw A èna mh kenì �nw fragmèno uposÔnolo tou dia-tetagmènou s¸matoc Σ. Lème ìti to α ∈ Σ eÐnai el�qisto �nw fr�gma touA an

• to α eÐnai �nw fr�gma tou A kai

• an α1 eÐnai �llo �nw fr�gma tou A tìte α ≤ α1.

(b) 'Estw A èna mh kenì k�tw fragmèno uposÔnolo tou diatetagmènou s¸matocΣ. Lème ìti to α ∈ Σ eÐnai mègisto k�tw fr�gma tou A an

• to α eÐnai k�tw fr�gma tou A kai

• an α1 eÐnai �llo k�tw fr�gma tou A tìte α ≥ α1.

Parat rhsh 1.3.4. To el�qisto �nw fr�gma tou A (an up�rqei) eÐnai mona-dikì. Apì ton orismì eÐnai fanerì ìti an α, α1 eÐnai dÔo el�qista �nw fr�gmatatou A tìte α ≤ α1 kai α1 ≤ α, dhlad α = α1. OmoÐwc, to mègisto k�tw fr�gmatou A (an up�rqei) eÐnai monadikì.

Sthn perÐptwsh pou up�rqoun, ja sumbolÐzoume to el�qisto �nw fr�gma touA me supA (to supremum tou A) kai to mègisto k�tw fr�gma tou A me inf A

(to infimum tou A). Ta inf A, supA mporeÐ na an koun na mhn an koun stosÔnolo A.

Orismìc 1.3.5. Lème ìti èna diatetagmèno s¸ma Σ ikanopoieÐ thn arq thcplhrìthtac an

K�je mh kenì kai �nw fragmèno uposÔnolo A tou Σ èqei el�qisto�nw fr�gma α ∈ Σ.

'Ena diatetagmèno s¸ma Σ pou ikanopoieÐ thn arq thc plhrìthtac lègetai pl -rwc diatetagmèno s¸ma.

H epìmenh Prìtash deÐqnei ìti to (Q, +, ·, <), me tic sun jeic pr�xeic kai thsun jh di�taxh, den ikanopoieÐ thn arq thc plhrìthtac.

Prìtash 1.3.6. To Q den eÐnai pl rwc diatetagmèno s¸ma: up�rqei mh kenì�nw fragmèno uposÔnolo A tou Q to opoÐo den èqei el�qisto �nw fr�gma.


A = {x ∈ Q : x > 0 kai x2 < 2}.

ParathroÔme pr¸ta ìti to A eÐnai mh kenì: èqoume 1 ∈ A (diìti 1 > 0 kai12 = 1 < 2). Qrhsimopoi¸ntac to gegonìc ìti an x, y eÐnai jetikoÐ rhtoÐ tìtex < y an kai mìno an x2 < y2 èqoume thn ex c:

1.3 Pragmatikoi arijmoi · 11

Parat rhsh: an gia k�poion jetikì rhtì y isqÔei y2 > 2 tìte o y

eÐnai �nw fr�gma tou A.

'Epetai ìti to A eÐnai �nw fragmèno: gia par�deigma, o 2 eÐnai �nw fr�gma tou A

afoÔ 2 > 0 kai 22 = 4 > 2.Upojètoume ìti to A èqei el�qisto �nw fr�gma, èstw a ∈ Q, kai ja katal -

xoume se �topo. AfoÔ den up�rqei rhtìc pou to tetr�gwno tou na isoÔtai me 2,anagkastik� ja isqÔei mÐa apì tic a2 > 2 a2 < 2 :(i) Upojètoume ìti a2 > 2. Ja broÔme 0 < ε < a ¸ste (a − ε)2 > 2. Tìte jaèqoume a − ε < a kai apì thn Parat rhsh, o a − ε ja eÐnai �nw fr�gma tou A,�topo.Epilog tou ε: Zht�me 0 < ε < a kai

(a− ε)2 = a2 − 2aε + ε2 > 2.

AfoÔ ε2 > 0, arkeÐ na exasfalÐsoume thn a2 − 2aε > 2, h opoÐa eÐnai isodÔnamhme thn

ε <a2 − 2

2a.

Parathr ste ìti o a2−22a eÐnai jetikìc rhtìc arijmìc. An loipìn epilèxoume

ε =12

min{

a,a2 − 2

2a

},

tìte èqoume breÐ rhtì ε pou ikanopoieÐ tic 0 < ε < a kai (a− ε)2 > 2.

(ii) Upojètoume ìti a2 < 2. Ja broÔme rhtì ε > 0 ¸ste (a + ε)2 < 2. Tìte jaèqoume a + ε > a kai a + ε ∈ A, �topo afoÔ o a eÐnai �nw fr�gma tou A.Epilog tou ε: Zht�me ε > 0 kai

(a + ε)2 = a2 + 2aε + ε2 < 2.

Ja epilèxoume ε ≤ 1 opìte ja isqÔei

a2 + 2aε + ε2 ≤ a2 + 2aε + ε = a2 + ε(2a + 1),

diìti ε2 ≤ ε. ArkeÐ loipìn na exasfalÐsoume thn a2 +ε(2a+1) < 2, h opoÐa eÐnaiisodÔnamh me thn

ε <2− a2

2a + 1.

Parathr ste ìti o 2−a2

2a+1 eÐnai jetikìc rhtìc arijmìc. An loipìn epilèxoume

ε =12

min{

1,2− a2

2a + 1

},

tìte èqoume breÐ rhtì ε > 0 pou ikanopoieÐ thn (a + ε)2 < 2.

Upojètontac ìti to A èqei el�qisto �nw fr�gma ton a ∈ Q apokleÐsame tica2 < 2, a2 = 2 kai a2 > 2. 'Ara, to A den èqei el�qisto �nw fr�gma (sto Q). 2

Parathr ste ìti to {el�qisto �nw fr�gma} tou sunìlou A sthn apìdeixhthc Prìtashc 1.3.6 eÐnai akrib¸c to shmeÐo thc eujeÐac to opoÐo ja antistoiqoÔsesto m koc thc upoteÐnousac tou orjogwnÐou trig¸nou me k�jetec pleurèc Ðsecme 1 (to opoÐo {leÐpei} apì to Q).

'Olh h doulei� pou ja k�noume se autì to m�jhma basÐzetai sto ex c je-¸rhma epèktashc (gia perissìterec leptomèreiec deÐte to Par�rthma autoÔ touKefalaÐou).

Je¸rhma 1.3.7. To diatetagmèno s¸ma (Q, +, ·, <) epekteÐnetai se èna pl rwcdiatetagmèno s¸ma (R, +, ·, <).

Deqìmaste dhlad ìti up�rqei èna pl rwc diatetagmèno s¸ma (R, +, ·, <) toopoÐo perièqei touc rhtoÔc (touc akeraÐouc kai touc fusikoÔc). To R eÐnai tosÔnolo twn pragmatik¸n arijm¸n. Oi pr�xeic + kai · sto R epekteÐnoun ticantÐstoiqec pr�xeic sto Q, ikanopoioÔn ta axi¸mata thc prìsjeshc, ta axi¸matatou pollaplasiasmoÔ kai thn epimeristik idiìthta. H di�taxh < sto R epekteÐneithn di�taxh sto Q kai ikanopoieÐ ta axi¸mata thc di�taxhc. Epiplèon, sto RisqÔei h arq thc plhrìthtac.

Arq thc plhrìthtac gia touc pragmatikoÔc arijmoÔc. K�je mhkenì, �nw fragmèno uposÔnolo A tou R èqei el�qisto �nw fr�gma α ∈ R.

MporeÐ kaneÐc na deÐxei ìti up�rqei {mìno èna} pl rwc diatetagmèno s¸ma (hepèktash mporeÐ na gÐnei me ènan ousiastik� trìpo). DÔo pl rwc diatetagmènas¸mata eÐnai isìmorfa (blèpe M. Spivak, Kef�laio 29).

T¸ra mporoÔme na deÐxoume ìti h exÐswsh x2 = 2 èqei lÔsh sto sÔnolo twnpragmatik¸n arijm¸n.

Prìtash 1.3.8. Up�rqei monadikìc jetikìc x ∈ R ¸ste x2 = 2.


A = {x ∈ R : x > 0 kai x2 < 2}.

ParathroÔme pr¸ta ìti to A eÐnai mh kenì: èqoume 1 ∈ A (diìti 1 > 0 kai12 = 1 < 2). Qrhsimopoi¸ntac to gegonìc ìti an x, y eÐnai jetikoÐ pragmatikoÐarijmoÐ tìte x < y an kai mìno an x2 < y2 èqoume thn ex c:

Parat rhsh: an gia k�poion jetikì pragmatikì y isqÔei y2 > 2 tìteo y eÐnai �nw fr�gma tou A.

'Epetai ìti to A eÐnai �nw fragmèno: gia par�deigma, o 2 eÐnai �nw fr�gma tou A

afoÔ 2 > 0 kai 22 = 4 > 2.Apì thn arq thc plhrìthtac, to A èqei el�qisto �nw fr�gma, èstw a ∈ R.

Profan¸c, a > 0. Ja deÐxoume ìti a2 = 2 apokleÐontac tic a2 > 2 kai a2 < 2 :

(i) Upojètoume ìti a2 > 2. Me to epiqeÐrhma thc apìdeixhc thc Prìtashc 1.3.6brÐskoume 0 < ε < a sto R ¸ste (a − ε)2 > 2. Tìte, a − ε < a kai apì thnParat rhsh, o a− ε eÐnai �nw fr�gma tou A, �topo.

1.4 Sunepeiec tou axiwmatoc thc plhrothtac · 13

(ii) Upojètoume ìti a2 < 2. Me to epiqeÐrhma thc apìdeixhc thc Prìtashc 1.3.6brÐskoume ε > 0 sto R ¸ste (a + ε)2 > 2. Tìte, a + ε > a kai a ∈ A, �topoafoÔ o a eÐnai �nw fr�gma tou A.

Anagkastik�, a2 = 2. H monadikìthta eÐnai apl : qrhsimopoi ste to gegonìcìti an x, y eÐnai jetikoÐ pragmatikoÐ arijmoÐ tìte x = y an kai mìno an x2 = y2.2

1.3bþ Qarakthrismìc tou supremum

H epìmenh Prìtash dÐnei ènan polÔ qr simo {ε�qarakthrismì} tou supremumenìc mh kenoÔ �nw fragmènou uposunìlou tou R.

Prìtash 1.3.9. 'Estw A mh kenì �nw fragmèno uposÔnolo tou R kai èstwα ∈ R. Tìte, α = sup A an kai mìno an isqÔoun ta ex c:

(a) To α eÐnai �nw fr�gma tou A,

(b) Gia k�je ε > 0 up�rqei x ∈ A ¸ste x > α− ε.

Apìdeixh. Upojètoume pr¸ta ìti α = sup A. Apì ton orismì tou supremum,ikanopoieÐtai to (a). Gia to (b), èstw ε > 0. An gia k�je x ∈ A Ðsque h x ≤ α−ε,tìte to α − ε ja tan �nw fr�gma tou A. Apì ton orismì tou supremum jaèprepe na èqoume

α ≤ α− ε, dhlad ε ≤ 0,

to opoÐo eÐnai �topo. 'Ara, gia to tuqìn ε > 0 up�rqei x ∈ A (to x exart�taibèbaia apì to ε) pou ikanopoieÐ thn x > α− ε.

AntÐstrofa, èstw α ∈ R pou ikanopoieÐ ta (a) kai (b). Eidikìtera, to A eÐnai�nw fragmèno. Ac upojèsoume ìti to α den eÐnai to supremum tou A. Tìte,up�rqei β < α to opoÐo eÐnai �nw fr�gma tou A. Jètoume ε = α− β > 0. Tìte,

x ≤ β = α− ε

gia k�je x ∈ A. Autì èrqetai se antÐfash me to (b). 2

'Askhsh 1.3.10. DeÐxte ìti k�je mh kenì k�tw fragmèno uposÔnolo A touR èqei mègisto k�tw fr�gma.

1.4 Sunèpeiec tou axi¸matoc thc plhrìthtac

Se aut thn par�grafo, qrhsimopoi¸ntac to axÐwma thc plhrìthtac, ja apodeÐ-xoume k�poiec basikèc idiìthtec tou sunìlou twn pragmatik¸n arijm¸n.

1.4aþ 'Uparxh n�ost c rÐzac

To diwnumikì an�ptugma. Gia k�je n ∈ N orÐzoume n! = 1 · 2 · · ·n (toginìmeno ìlwn twn fusik¸n apì 1 wc n). SumfwnoÔme ìti 0! = 1. Parathr steìti n! = n(n− 1)! gia k�je n ∈ N.

An 0 ≤ k ≤ n orÐzoume(

n

k

)=

n!k!(n− k)!

=n(n− 1) · · · (n− k + 1)

k!.

Parathr ste ìti (n

0

)=

(n

n

)= 1

gia k�je n = 0, 1, 2, . . ..

L mma 1.4.1 (trÐgwno tou Pascal). An 1 ≤ k < n tìte(

n

k

)=

(n− 1

k

)+

(n− 1k − 1

).

Apìdeixh. Me b�sh touc orismoÔc pou d¸same, mporoÔme na gr�youme(

n− 1k

)+

(n− 1k − 1

)=

(n− 1)!k!(n− k − 1)!

+(n− 1)!

(k − 1)!(n− k)!

=(n− 1)!(n− k)

k!(n− k − 1)!(n− k)+

(n− 1)!k(k − 1)!k(n− k)!

=(n− 1)!(n− k)

k!(n− k)!+

(n− 1)!kk!(n− k)!

=(n− 1)![(n− k) + k]

k!(n− k)!=

(n− 1)!nk!(n− k)!

=(

n

k

),

dhlad to zhtoÔmeno. 2

Sumbolismìc. An a0, a1, . . . , an ∈ R orÐzoume

n∑

k=0

ak = a0 + a1 + · · ·+ an.

Parathr ste ìti to �jroisma a0 + a1 + · · ·+ an mporeÐ isodÔnama na grafteÐ wcex c:

n∑

k=0

ak =n∑

m=0

am =n+1∑s=1

as−1.

H pr¸th isìthta isqÔei giatÐ all�xame (apl¸c) to {ìnoma} thc metablht c apìk se m. H deÔterh giatÐ k�name (apl¸c) thn {allag metablht c} s = m + 1.

Prìtash 1.4.2 (diwnumikì an�ptugma). Gia k�je a, b ∈ R kai k�jen ∈ N isqÔei

(a + b)n =n∑

k=0

(n

k

)an−kbk.

Apìdeixh. Me epagwg : gia n = 1 h zhtoÔmenh isìthta gr�fetai

a + b =(

10

)a1b0 +

(11

)a0b1,

h opoÐa isqÔei: parathr ste ìti(10

)=

(11

)= 1, a0 = b0 = 1, a1 = a kai b1 = b.

Upojètoume ìti

(a + b)n =n∑

k=0

(n

k

)an−kbk

kai deÐqnoume ìti

(a + b)n+1 =n+1∑

k=0

(n + 1

k

)an+1−kbk.

Pr�gmati,

(a + b)n+1 = (a + b)(a + b)n = (a + b)

[n∑

k=0

(n

k

)an−kbk

]

= a ·n∑

k=0

(n

k

)an−kbk + b ·

n∑

k=0

(n

k

)an−kbk

=n∑

k=0

(n

k

)an+1−kbk +

n∑m=0

(n

m

)an−mbm+1

= an+1 +n∑

k=1

(n

k

)an+1−kbk +

n−1∑m=0

(n

m

)an−mbm+1 + bn+1

= an+1 +n∑

k=1

(n

k

)an+1−kbk +

n∑

k=1

(n

k − 1

)an−(k−1)bk + bn+1

= an+1 +n∑

k=1

[(n

k

)+

(n

k − 1

)]an+1−kbk + bn+1.

Apì to L mma èqoume(n+1

k

)=

(nk

)+

(n

k−1

), �ra

(a + b)n+1 = an+1 +n∑

k=1

(n + 1

k

)an+1−kbk + bn+1 =

n+1∑

k=0

(n + 1

k

)an+1−kbk.

Autì oloklhr¸nei to epagwgikì b ma kai thn apìdeixh. 2

Je¸rhma 1.4.3 (Ôparxh n�ost c rÐzac). 'Estw ρ ∈ R, ρ > 0 kai èstwn ∈ N. Up�rqei monadikìc x > 0 sto R ¸ste xn = ρ.

[O x sumbolÐzetai me n√

ρ ρ1/n. Profan¸c mac endiafèrei mìno h perÐptwshn ≥ 2.]

Apìdeixh. Upojètoume pr¸ta ìti ρ > 1. JewroÔme to sÔnolo

A = {y ∈ R : y > 0 kai yn < ρ}.

To A eÐnai mh kenì: èqoume 1 ∈ A. ParathroÔme ìti k�je jetikìc pragmatikìcarijmìc α me thn idiìthta αn > ρ eÐnai �nw fr�gma tou A: an y ∈ A tìteyn < ρ < αn kai, afoÔ y, α > 0, sumperaÐnoume ìti y < α. 'Ena tètoio �nwfr�gma tou A eÐnai o ρ: apì thn ρ > 1 èpetai ìti ρn > ρ.AfoÔ to A eÐnai mh kenì kai �nw fragmèno, apì to axÐwma thc plhrìthtac, up�rqeio x = sup A. Ja deÐxoume ìti xn = ρ.(a) 'Estw ìti xn < ρ. Ja broÔme ε > 0 ¸ste (x + ε)n < ρ, dhlad x + ε ∈ A

(�topo, giatÐ o x èqei upotejeÐ �nw fr�gma tou A).An upojèsoume apì thn arq ìti 0 < ε ≤ 1, èqoume

(x + ε)n = xn +n∑

k=1

(n

k

)xn−kεk = xn + ε

[n∑

k=1

(n

k

)xn−kεk−1

]

≤ xn + ε

[n∑

k=1

(n

k

)xn−k

].

Ja èqoume loipìn (x+ ε)n < ρ an epilèxoume 0 < ε < ρ−xn

∑nk=1 (n

k)xn−k. Epilègoume

ε =12

min

{1,

ρ− xn

∑nk=1

(nk

)xn−k

}.

O ε eÐnai jetikìc pragmatikìc arijmìc (diìti ρ− xn > 0 kai∑n

k=1

(nk

)xn−k > 0)

kai (x + ε)n < ρ.(b) 'Estw ìti xn > ρ. Ja broÔme 0 < ε < x ¸ste (x − ε)n > ρ (�topo, giatÐtìte o x− ε ja tan �nw fr�gma tou A mikrìtero apì to sup A).

Gia k�je 0 < ε < x èqoume

(x− ε)n = xn +n∑

k=1

(n

k

)xn−k(−1)kεk = xn − ε

[n∑

k=1

(n

k

)xn−k(−1)k−1εk−1

]

≥ xn − ε

[n∑

k=1

(n

k

)xn−k

].

Ja èqoume loipìn (x− ε)n > ρ an epilèxoume 0 < ε < xn−ρ∑nk=1 (n

k)xn−k. Epilègoume

ε =12

min

{x,

xn − ρ∑nk=1

(nk

)xn−k

}.

O ε eÐnai jetikìc pragmatikìc arijmìc (diìti xn − ρ > 0 kai∑n

k=1

(nk

)xn−k > 0)

kai gia ton jetikì pragmatikì arijmì x− ε isqÔei (x− ε)n > ρ.ApokleÐsame tic xn < ρ kai xn > ρ. Sunep¸c, xn = ρ. H monadikìthta eÐnaiapl : parathr ste ìti an 0 < x1 < x2 tìte xn

1 < xn2 gia k�je n ∈ N.

An 0 < ρ < 1 èqoume 1ρ > 1 kai, apì to prohgoÔmeno b ma, up�rqei monadikìc

x > 0 ¸ste xn = 1ρ . JewroÔme ton 1

x . Tìte,(

1x

)n

=1xn

= ρ.

Tèloc, an ρ = 1 jewroÔme ton x = 1. 2

1.4bþ Arqim deia idiìthta

Pr¸to mac b ma eÐnai na deÐxoume ìti to N den eÐnai �nw fragmèno uposÔnolotou R:

Je¸rhma 1.4.4. To sÔnolo N twn fusik¸n arijm¸n den eÐnai �nw fragmènouposÔnolo tou R.

Apìdeixh. Me apagwg se �topo. Upojètoume ìti to sÔnolo N eÐnai �nw frag-mèno. Apì to axÐwma thc plhrìthtac to N èqei el�qisto �nw fr�gma: èstwβ = supN. Tìte β − 1 < β, �ra o β − 1 den eÐnai �nw fr�gma tou N. MporoÔmeloipìn na broÔme n ∈ N me n > β−1. 'Epetai ìti n+1 > β, �topo afoÔ n+1 ∈ Nkai o β eÐnai �nw fr�gma tou N. 2

IsodÔnamoi trìpoi diatÔpwshc thc Ðdiac arq c eÐnai oi ex c.

Je¸rhma 1.4.5 (Arqim deia idiìthta twn pragmatik¸n). 'Estw ε

kai a dÔo pragmatikoÐ arijmoÐ me ε > 0. Up�rqei n ∈ N ¸ste nε > a.

Apìdeixh. Apì to Je¸rhma 1.4.4 o aε den eÐnai �nw fr�gma tou N. Sunep¸c,

up�rqei n ∈ N ¸ste n > aε . AfoÔ ε > 0, èpetai ìti nε > a. 2

Je¸rhma 1.4.6. 'Estw ε > 0. Up�rqei n ∈ N ¸ste 0 < 1n < ε.

Apìdeixh. Apì to Je¸rhma 1.4.4 o 1ε den eÐnai �nw fr�gma tou N. Sunep¸c,

up�rqei n ∈ N ¸ste n > 1ε . AfoÔ ε > 0, èpetai ìti 1

n < ε. 2

1.4gþ 'Uparxh akeraÐou mèrouc

DeÐqnoume pr¸ta thn ex c epèktash thc arq c tou elaqÐstou (parathr ste ìtiqrhsimopoioÔme thn Arqim deia idiìthta twn pragmatik¸n arijm¸n).

Prìtash 1.4.7. K�je mh kenì uposÔnolo tou Z pou eÐnai k�tw fragmèno èqeiel�qisto stoiqeÐo.

Apìdeixh. 'Estw A 6= ∅ k�tw fragmèno uposÔnolo tou Z. Up�rqei x ∈ R ¸stex ≤ a gia k�je a ∈ A. Apì thn Arqim deia idiìthta twn pragmatik¸n arium¸n,up�rqei n ∈ N me n > −x, dhlad −n < x ≤ a gia k�je a ∈ A. Up�rqei dhlad m ∈ Z pou eÐnai k�tw fr�gma tou A (p�rte m = −n), kai m�lista {gn sio} methn ènnoia ìti

m ∈ Z kai m < a gia k�je a ∈ A.

JewroÔme to sÔnolo B = {a −m : a ∈ A} ⊆ N. To B èqei el�qisto stoiqeÐo,to opoÐo onom�zoume β. Dhlad ,

β = a0 −m gia k�poio a0 ∈ A kai β ≤ a−m gia k�je a ∈ A.

Tìte, o a0 eÐnai to el�qisto stoiqeÐo tou A: profan¸c a0 ∈ A, kai gia k�jea ∈ A èqoume a0 −m ≤ a−m =⇒ a0 ≤ a. 2

Me an�logo trìpo mporeÐte na deÐxete ìti k�je mh kenì kai �nw fragmènosÔnolo akeraÐwn arijm¸n èqei mègisto stoiqeÐo. DÐnoume mia deÔterh apìdeixh

autoÔ tou duðkoÔ isqurismoÔ, qrhsimopoi¸ntac apeujeÐac aut to for� to axÐwmathc plhrìthtac.

DeÔterh apìdeixh. 'Estw A èna mh kenì kai �nw fragmèno uposÔnolo tou Z.Apì to axÐwma thc plhrìthtac, up�rqei to a = sup A ∈ R. Ja deÐxoume ìtia ∈ A: apì ton qarakthrismì tou supremum, up�rqei x ∈ A ¸ste a−1 < x ≤ a.An a /∈ A, tìte x < a. Autì shmaÐnei ìti o x den eÐnai �nw fr�gma tou A,opìte, efarmìzontac p�li ton qarakthrismì tou supremum, brÐskoume y ∈ A

¸ste a− 1 < x < y < a. 'Epetai ìti 0 < y − x < 1. Autì eÐnai �topo diìti oi x

kai y eÐnai akèraioi. 2

Je¸rhma 1.4.8 (Ôparxh akeraÐou mèrouc). Gia k�je x ∈ R up�rqeimonadikìc akèraioc m ∈ Z me thn idiìthta

m ≤ x < m + 1.

Apìdeixh. To sÔnolo A = {n ∈ Z : n > x} eÐnai mh kenì (apì thn Arqim deiaidiìthta) kai k�tw fragmèno apì to x. Apì thn Prìtash 1.4.7, to A èqei el�qistostoiqeÐo : ac to poÔme n0. AfoÔ n0 − 1 /∈ A, èqoume n0 − 1 ≤ x. Jètoumem = n0 − 1. EÐdame ìti m ≤ x. EpÐshc n0 ∈ A, dhlad m + 1 > x. 'Ara,

m ≤ x < m + 1.

Gia th monadikìthta ac upojèsoume ìti

m ≤ x < m + 1 kai m1 ≤ x < m1 + 1

ìpou m,m1 ∈ Z. 'Eqoume m < m1 + 1 �ra m ≤ m1, kai m1 < m + 1 �ram1 ≤ m. Sunep¸c, m = m1. 2

Orismìc 1.4.9. O akèraioc m pou mac dÐnei to prohgoÔmeno je¸rhma (kaio opoÐoc exart�tai k�je for� apì ton x) lègetai akèraio mèroc tou x, kaisumbolÐzetai me [x]. Dhlad , o [x] prosdiorÐzetai apì tic

[x] ∈ Z kai [x] ≤ x < [x] + 1.

Gia par�deigma, [2.7] = 2, [−2.7] = −3.

1.4dþ Puknìthta twn rht¸n kai twn arr twn stouc pragmati-koÔc arijmoÔc

H Ôparxh tou akeraÐou mèrouc kai h Arqim deia idiìthta twn pragmatik¸n arijm¸nmac exasfalÐzoun thn puknìthta tou Q sto R: an�mesa se opoiousd pote dÔopragmatikoÔc arijmoÔc mporoÔme na broÔme ènan rhtì.

Je¸rhma 1.4.10. An x, y ∈ R kai x < y, tìte up�rqei rhtìc q me thn idiìthta

x < q < y.

1.5 Orismoi kai sumbolismoc · 19

Apìdeixh. 'Eqoume y − x > 0 kai apì thn Arqim deia idiìthta up�rqei fusikìcn ∈ N ¸ste n(y − x) > 1, dhlad

nx + 1 < ny.

Tìte,nx < [nx] + 1 ≤ nx + 1 < ny,

dhlad

x <[nx] + 1

n< y.

AfoÔ o q = [nx]+1n eÐnai rhtìc, èqoume to zhtoÔmeno. 2

Orismìc 1.4.11. Sthn §1.3 eÐdame ìti to Q eÐnai gn sio uposÔnolo tou R:up�rqei pragmatikìc arijmìc x > 0 me x2 = 2, kai o x den eÐnai rhtìc. K�jepragmatikìc arijmìc pou den eÐnai rhtìc lègetai �rrhtoc.

Je¸rhma 1.4.12. Oi �rrhtoi eÐnai puknoÐ sto R: an x, y ∈ R kai x < y, tìteup�rqei α �rrhtoc me x < α < y.

Apìdeixh. 'Eqoume x < y kai√

2 > 0, �ra x/√

2 < y/√

2. Apì to Je¸rhma1.4.10, up�rqei rhtìc q me

x√2

< q <y√2.

MporoÔme na upojèsoume ìti q 6= 0 (alli¸c ja paÐrname �llon rhtì q′ an�mesaston x/

√2 kai ston q = 0). 'Epetai ìti o α := q

√2 eÐnai �rrhtoc (exhg ste

giatÐ) kaix < α = q

√2 < y.

1.5 OrismoÐ kai sumbolismìc

1.5aþ Apìluth tim

Orismìc 1.5.1 (apìluth tim ). Gia k�je a ∈ R jètoume

|a| =

a an a ≥ 0,

−a an a < 0.

O |a| lègetai apìluth tim tou a. Jewr¸ntac ton a san shmeÐo thc eujeÐ-ac, skeftìmaste thn apìluth tim tou san thn {apìstash} tou a apì to 0.Parathr ste ìti | − a| = |a| kai |a| ≥ 0 gia k�je a ∈ R.

Prìtash 1.5.2. Gia k�je a ∈ R kai ρ ≥ 0 isqÔei

|a| ≤ ρ an kai mìno an − ρ ≤ a ≤ ρ.

Apìdeixh. DiakrÐnete peript¸seic: a ≥ 0 kai a < 0. 2

Prìtash 1.5.3 (trigwnik anisìthta). Gia k�je a, b ∈ R,

|a + b| ≤ |a|+ |b|.

EpÐshc,| |a| − |b| | ≤ |a− b| kai | |a| − |b| | ≤ |a + b| .

Apìdeixh. Apì thn Prìtash 1.5.2 èqoume −|a| ≤ a ≤ |a| kai −|b| ≤ b ≤ |b|.Sunep¸c,

−(|a|+ |b|) ≤ a + b ≤ |a|+ |b|.Qrhsimopoi¸ntac p�li thn Prìtash 1.5.2 sumperaÐnoume ìti |a + b| ≤ |a|+ |b|.

Gia th deÔterh anisìthta gr�foume

|a| = |(a− b) + b| ≤ |a− b|+ |b|,

opìte |a| − |b| ≤ |a− b|. Me ton Ðdio trìpo blèpoume ìti

|b| = |(b− a) + a| ≤ |b− a|+ |a| = |a− b|+ |a|,

�ra |b| − |a| ≤ |a− b|. AfoÔ

−|a− b| ≤ |a| − |b| ≤ |a− b|,

h Prìtash 1.5.2 deÐqnei ìti | |a| − |b| | ≤ |a− b|.An sthn teleutaÐa anisìthta antikatast soume ton b me ton −b, blèpoume

ìti | |a| − |b| | ≤ |a + b|. 2

1.5bþ To sÔnolo twn epektetamènwn pragmatik¸n arijm¸n

EpekteÐnoume to sÔnolo R twn pragmatik¸n arijm¸n me duo akìma stoiqeÐa,to +∞ kai to −∞. To sÔnolo R = R ∪ {+∞,−∞} eÐnai to sÔnolo twn epe-ktetamènwn pragmatik¸n arijm¸n. EpekteÐnoume th di�taxh kai tic pr�xeic stoR wc ex c:(a) OrÐzoume −∞ < a kai a < +∞ gia k�je a ∈ R.(b) Gia k�je a ∈ R orÐzoume

a + (+∞) = (+∞) + a = a− (−∞) = +∞a + (−∞) = (−∞) + a = a− (+∞) = −∞.

(g) An a > 0 orÐzoume

a · (+∞) = (+∞) · a = +∞a · (−∞) = (−∞) · a = −∞.

(d) An a < 0 orÐzoume

a · (+∞) = (+∞) · a = −∞a · (−∞) = (−∞) · a = +∞.

1.5 Orismoi kai sumbolismoc · 21

(e) EpÐshc, orÐzoume

(+∞) + (+∞) = +∞ (−∞) + (−∞) = −∞(+∞) · (+∞) = +∞ (−∞) · (−∞) = +∞

kai(+∞) · (−∞) = (−∞) · (+∞) = −∞.

(st) Den orÐzontai oi parast�seic

(+∞) + (−∞), (−∞) + (+∞), 0 · (+∞), (+∞) · 0, 0 · (−∞), (−∞) · 0

kai+∞+∞ ,

+∞−∞ ,

−∞+∞ ,

−∞−∞ .

Tèloc, an èna mh kenì sÔnolo A ⊆ R den eÐnai �nw fragmèno orÐzoume supA =+∞, en¸ an den eÐnai k�tw fragmèno orÐzoume inf A = −∞.

1.5gþ Diast mata

Orismìc 1.5.4. 'Estw a, b ∈ R me a < b. OrÐzoume

[a, b] = {x ∈ R : a ≤ x ≤ b}(a, b) = {x ∈ R : a < x < b}[a, b) = {x ∈ R : a ≤ x < b}(a, b] = {x ∈ R : a < x ≤ b}

[a,+∞) = {x ∈ R : x ≥ a}(a,+∞) = {x ∈ R : x > a}(−∞, b] = {x ∈ R : x ≤ b}(−∞, b) = {x ∈ R : x < b}.

Ta uposÔnola aut� tou sunìlou twn pragmatik¸n arijm¸n lègontai diast -mata.

Sto epìmeno L mma perigr�foume ta shmeÐa tou kleistoÔ diast matoc [a, b].

L mma 1.5.5. An a < b sto R tìte

[a, b] = {(1− t)a + tb : 0 ≤ t ≤ 1}.

Eidikìtera, gia k�je x ∈ [a, b] èqoume

x =b− x

b− aa +

x− a

b− ab.

Apìdeixh. EÔkola elègqoume ìti, gia k�je t ∈ [0, 1] isqÔei

a ≤ (1− t)a + tb = a + t(b− a) ≤ b,


dhlad {(1− t)a + tb : 0 ≤ t ≤ 1} ⊆ [a, b].AntÐstrofa, k�je x ∈ [a, b] gr�fetai sth morf

x =b− x

b− aa +

x− a

b− ab.

Parathr¸ntac ìti t := (x−a)/(b−a) ∈ [0, 1] kai 1−t = (b−x)/(b−a), blèpoumeìti [a, b] ⊆ {(1− t)a + tb : 0 ≤ t ≤ 1}. 2

Ta shmeÐa (1− t)a + tb tou [a, b] lègontai kurtoÐ sunduasmoÐ twn a kaib. To mèso tou [a, b] eÐnai to

m = (1− 12 )a + 1

2b =a + b

2.

1.6 Anisìthtec

Se aut thn par�grafo deÐqnoume me epagwg dÔo basikèc anisìthtec: thn ani-sìthta tou Bernoulli kai thn anisìthta arijmhtikoÔ�gewmetrikoÔ mèsou. 'Allecbasikèc anisìthtec emfanÐzontai stic Ask seic.

Prìtash 1.6.1 (anisìthta tou Bernoulli). An x > −1 tìte

(1 + x)n ≥ 1 + nx gia k�je n ∈ N.

Apìdeixh. Gia n = 1 h anisìthta isqÔei wc isìthta: 1 + x = 1 + x. DeÐqnoumeto epagwgikì b ma:

Upojètoume ìti (1+x)n ≥ 1+nx. AfoÔ 1+x > 0, èqoume (1+x)(1+x)n ≥(1 + x)(1 + nx). 'Ara,

(1 + x)n+1 ≥ (1 + x)(1 + nx) = 1 + (n + 1)x + nx2 ≥ 1 + (n + 1)x. 2

Parat rhsh. An x > 0, mporoÔme na deÐxoume thn anisìthta tou Bernoulliqrhsimopoi¸ntac to diwnumikì an�ptugma: gia k�je n ≥ 2 èqoume

(1 + x)n =n∑

k=0

(n

k

)1n−kxk = 1 + nx +

n∑

k=2

(n

k

)xn−k > 1 + nx,

afoÔ ìloi oi prosjetèoi sto∑n

k=2

(nk

)xn−k eÐnai jetikoÐ. OmoÐwc, an n ≥ 3

paÐrnoume thn isqurìterh anisìthta

(1 + x)n > 1 + nx +(

n

2

)x2 = 1 + nx +

n(n− 1)2

x2.

Prìtash 1.6.2 (anisìthta arijmhtikoÔ�gewmetrikoÔ mèsou). 'E-stw n ∈ N. An a1, . . . , an eÐnai jetikoÐ pragmatikoÐ arijmoÐ, tìte

a1 + a2 + · · ·+ an

n≥ n√

a1a2 · · · an.

1.6 Anisothtec · 23

Apìdeixh. Jètoume m = n√

a1a2 · · · an kai orÐzoume bk = ak

m , k = 1, . . . , n.ParathroÔme ìti oi bk eÐnai jetikoÐ pragmatikoÐ arijmoÐ me ginìmeno

b1 · · · bn =a1

m· · · an

m=

a1 · · · an

mn= 1.

EpÐshc, h zhtoÔmenh anisìthta paÐrnei th morf

b1 + · · ·+ bn ≥ n.

ArkeÐ loipìn na deÐxoume thn akìloujh Prìtash.

Prìtash 1.6.3. 'Estw n ∈ N. An b1, . . . , bn eÐnai jetikoÐ pragmatikoÐ arijmoÐme ginìmeno b1 · · · bn = 1, tìte b1 + · · ·+ bn ≥ n.

Apìdeixh. Me epagwg wc proc to pl joc twn bk: an n = 1 tìte èqoume ènanmìno arijmì, ton b1 = 1. Sunep¸c, h anisìthta eÐnai tetrimmènh: 1 ≥ 1.

Upojètoume ìti gia k�je m-�da jetik¸n arijm¸n x1, . . . , xm me ginìmenox1 · · ·xm = 1 isqÔei h anisìthta

x1 + · · ·+ xm ≥ m,

kai deÐqnoume ìti an b1, · · · , bm+1 eÐnai (m + 1) jetikoÐ pragmatikoÐ arijmoÐ meginìmeno b1 · · · bm+1 = 1 tìte

b1 + · · ·+ bm+1 ≥ m + 1.

MporoÔme na upojèsoume ìti b1 ≤ b2 ≤ · · · ≤ bm+1. ParathroÔme ìti, an b1 =b2 = · · · = bm+1 = 1 tìte h anisìthta isqÔei san isìthta. An ìqi, anagkastik�èqoume b1 < 1 < bm+1 (exhg ste giatÐ).

JewroÔme thn m-�da jetik¸n arijm¸n

x1 = b1bm+1 , x2 = b2, . . . , xm = bm.

AfoÔ x1 · · ·xm = b1 · · · bm+1 = 1, apì thn epagwgik upìjesh paÐrnoume

(b1bm+1) + b2 + · · ·+ bm = x1 + · · ·+ xm ≥ m.

'Omwc, apì thn b1 < 1 < bm+1 èpetai ìti (bm+1−1)(1−b1) > 0 dhlad b1+bm+1 >

1 + bm+1b1. 'Ara,

b1 + bm+1 + b2 + · · ·+ bm > 1 + b1bm+1 + b2 + · · ·+ bm ≥ 1 + m.

'Eqoume loipìn deÐxei to epagwgikì b ma. 2

Parat rhsh. An oi arijmoÐ a1, · · · , an eÐnai ìloi Ðsoi tìte h anisìthta arijmhtikoÔ�gewmetrikoÔ mèsou isqÔei wc isìthta. An oi arijmoÐ a1, · · · , an den eÐnai ìloi Ðsoi,tìte h apìdeixh pou prohg jhke deÐqnei ìti h anisìthta eÐnai gn sia (exhg stegiatÐ). Dhlad : sthn anisìthta arijmhtikoÔ�gewmetrikoÔ mèsou isqÔei isìthtaan kai mìnon an a1 = · · · = an.

1.7 *Par�rthma: Tomèc Dedekind

Upojètoume ed¸ ìti to sÔnolo Q twn rht¸n arijm¸n èqei oristeÐ, kai jew-roÔme ìlec tic idiìthtec tou gnwstèc. Ja perigr�youme thn kataskeu tou Rmèsw twn tom¸n Dedekind. Ta stoiqeÐa tou R ja eÐnai k�poia uposÔnola touQ, oi legìmenec tomèc. H idèa pÐsw apì ton orismì touc eÐnai ìti k�je pragma-tikìc arijmìc prosdiorÐzetai apì to sÔnolo twn rht¸n pou eÐnai mikrìteroÐ tou:an x ∈ R kai an orÐsoume Ax = {q ∈ Q : q < x}, tìte x = sup Ax.

Orismìc 1.7.1. 'Ena uposÔnolo α tou Q lègetai tom an ikanopoieÐ ta ex c:

• α 6= ∅, α 6= Q.

• an p ∈ α, q ∈ Q kai q < p, tìte q ∈ α.

• an p ∈ α, up�rqei q ∈ α ¸ste p < q.

H trÐth idiìthta mac lèei ìti mia tom α den èqei mègisto stoiqeÐo. H deÔterhèqei tic ex c �mesec sunèpeiec pou ja fanoÔn qr simec:

• an p ∈ α kai q /∈ α, tìte p < q.

• an r /∈ α kai r < s, tìte s /∈ α.

ShmeÐwsh. Se ìlh aut thn par�grafo qrhsimopoioÔme ta ellhnik� gr�mmataα, β, γ gia tomèc (=mellontikoÔc pragmatikoÔc arijmoÔc) kai ta latinik� p, q, r, s

gia rhtoÔc arijmoÔc.

B ma 1: OrÐzoume R = {α ⊆ Q : to α eÐnai tom }. Autì ja eÐnai telik� tosÔnolo twn pragmatik¸n arijm¸n.

B ma 2: Pr¸ta orÐzoume th di�taxh sto R. An α, β eÐnai dÔo tomèc, tìte

α < β ⇐⇒ to α eÐnai gn sio uposÔnolo tou β.

'Askhsh. DeÐxte ìti an α, β eÐnai tomèc, tìte isqÔei akrib¸c mÐa apì tic α < β,α = β, β < α.

B ma 3: To (R, <) ikanopoieÐ to axÐwma thc plhrìthtac. Dhlad , an A eÐnaimh kenì uposÔnolo tou R kai up�rqei tom β ∈ R ¸ste α ≤ β gia k�je α ∈ A,tìte to A èqei el�qisto �nw fr�gma.

Apìdeixh. OrÐzoume γ thn ènwsh ìlwn twn stoiqeÐwn tou A. Dhlad ,

γ = {q ∈ Q : ∃α ∈ A me q ∈ α}.

Ja deÐxoume ìti γ = sup A.(a) To γ eÐnai tom : Pr¸ton, γ 6= ∅: afoÔ A 6= ∅, up�rqei α0 ∈ A. AfoÔ α0 6= ∅,up�rqei q ∈ α0. Tìte, q ∈ γ. Prèpei epÐshc na deÐxoume ìti γ 6= Q: Up�rqeiq ∈ Q me q /∈ β. An α ∈ A, tìte α ≤ β, �ra q /∈ α. Epomènwc, q /∈ ∪{α : α ∈ A}dhlad q /∈ γ. 'Ara, to γ ikanopoieÐ thn pr¸th sunj kh tou orismoÔ thc tom c.

1.7 *Pararthma: Tomec Dedekind · 25

Gia th deÔterh, èstw p ∈ γ kai q ∈ Q me q < p. Up�rqei α ∈ A me p ∈ α kaiq < p, �ra q ∈ α. AfoÔ α ⊆ γ, èpetai ìti q ∈ γ.

Gia thn trÐth, èstw p ∈ γ. Up�rqei α ∈ A me p ∈ α. AfoÔ to α eÐnai tom ,up�rqei q ∈ α me p < q. Tìte, q ∈ γ kai p < q.(b) To γ eÐnai �nw fr�gma tou A: An α ∈ A, tìte α ⊆ γ dhlad α ≤ γ.(g) To γ eÐnai to el�qisto �nw fr�gma tou A: 'Estw β1 ∈ R �nw fr�gma tou A.Tìte β1 ≥ α gia k�je α ∈ A, dhlad β1 ⊇ α gia k�je α ∈ A, dhlad

β1 ⊇⋃{α : α ∈ A} = γ,

dhlad β1 ≥ γ. 2

B ma 4: OrÐzoume mia pr�xh + (prìsjesh) sto R wc ex c: an α, β ∈ R, tìteα + β = {p + q : p ∈ α, q ∈ β}.

(a) DeÐqnoume ìti to α+β eÐnai tom , kai eÔkola epalhjeÔoume ìti α+β = β+α

kai α + (β + γ) = (α + β) + γ gia k�je α, β, γ ∈ R.(b) OrÐzoume 0∗ = {q ∈ Q : q < 0} kai deÐqnoume ìti to 0∗ ∈ R kai eÐnai tooudètero stoiqeÐo thc prìsjeshc: α + 0∗ = 0∗ + α = α gia k�je α ∈ R.(g) An α ∈ R, to −α orÐzetai wc ex c:

−α = {q ∈ Q : up�rqei r ∈ Q, r > 0 me − q − r /∈ α}.DeÐxte ìti −α ∈ R kai α + (−α) = (−α) + α = 0∗.'Epetai ìti h pr�xh + sto R ikanopoieÐ ta axi¸mata thc prìsjeshc. 2

B ma 5: To sÔnolo Θ twn jetik¸n stoiqeÐwn tou R orÐzetai t¸ra me fusiolo-gikì trìpo:

α ∈ Θ ⇐⇒ 0∗ < α.

DeÐxte ìti an α ∈ R, tìte isqÔei akrib¸c mÐa apì tic α ∈ Θ, α = 0∗, −α ∈ Θ.

B ma 6: OrÐzoume mia pr�xh pollaplasiasmoÔ, pr¸ta gia α, β ∈ Θ: An α > 0∗

kai β > 0∗, jètoume

αβ = {q ∈ Q : up�rqoun r ∈ α, s ∈ β, r > 0, s > 0 me q ≤ rs}.

(a) DeÐqnoume ìti to αβ eÐnai tom kai αβ = βα, α(βγ) = (αβ)γ an α, β, γ ∈ Θ.(b) OrÐzoume 1∗ = {q ∈ Q : q < 1}. Tìte, α1∗ = 1∗α = α gia k�je α ∈ Θ.(g) An α ∈ Θ, o antÐstrofoc α−1 tou α orÐzetai apì thn:

α−1 = {q ∈ Q : q ≤ 0 q > 0 kai up�rqei r ∈ Q, r > 1 me (qr)−1 /∈ α}.DeÐxte ìti α−1 ∈ Θ kai αα−1 = α−1α = 1∗.Oloklhr¸noume ton orismì tou pollaplasiasmoÔ jètontac

αβ = (−α)(−β), an α, β < 0∗

αβ = −[(−α)β], an α < 0∗, β > 0∗

αβ = −[α(−β)], an α > 0∗, β < 0∗,

kaiα0∗ = 0∗α = 0∗.

MporoÔme t¸ra na doÔme ìti ikanopoioÔntai ìla ta axi¸mata tou pollaplasia-smoÔ, kaj¸c kai h epimeristik idiìthta tou pollaplasiasmoÔ wc proc thn prì-sjesh. Den ja mpoÔme se perissìterec leptomèreiec (an jèlete sumbouleuteÐteton M. Spivak, Kef�laio 28).

{To R me b�sh thn parap�nw kataskeu eÐnai èna pl rwc diatetag-mèno s¸ma.}

B ma 7: An q ∈ Q orÐzoume q∗ = {r ∈ Q : r < q}. K�je q∗ eÐnai tom , dhlad q∗ ∈ R. EÔkola deÐqnoume ìti:(a) an p, q ∈ Q, tìte p∗ + q∗ = (p + q)∗.(b) an p, q ∈ Q, tìte p∗q∗ = (pq)∗.(g) an p, q ∈ Q, tìte p∗ < q∗ an kai mìno an p < q.

Epomènwc, h apeikìnish I : Q → R me I(q) = q∗ diathreÐ tic pr�xeic thcprìsjeshc kai tou pollaplasiasmoÔ, kaj¸c kai th di�taxh. MporoÔme loipìn nablèpoume toQ san èna diatetagmèno upos¸ma tou R mèsw thc taÔtishcQ←→ Q∗

(ìpou Q∗ = {q∗ : q ∈ Q} ⊂ R).

1.8 Ask seic

A. Erwt seic katanìhshc

Exet�ste an oi parak�tw prot�seic eÐnai alhjeÐc yeudeÐc (aitiolog ste pl rwcthn ap�nthsh sac).

1. 'Estw A mh kenì, �nw fragmèno uposÔnolo tou R. Gia k�je x ∈ A èqoumex ≤ supA.

2. 'Estw A mh kenì, �nw fragmèno uposÔnolo tou R. O x ∈ R eÐnai �nw fr�gmatou A an kai mìno an supA ≤ x.

3. An to A eÐnai mh kenì kai �nw fragmèno uposÔnolo tou R tìte sup A ∈ A.

4. An A eÐnai èna mh kenì kai �nw fragmèno uposÔnolo tou Z tìte sup A ∈ A.

5. An a = sup A kai ε > 0, tìte up�rqei x ∈ A me a− ε < x ≤ a.

6. An a = sup A kai ε > 0, tìte up�rqei x ∈ A me a− ε < x < a.

7. An to A eÐnai mh kenì kai supA − inf A = 1 tìte up�rqoun x, y ∈ A ¸stex− y = 1.

8. Gia k�je x, y ∈ R me x < y up�rqoun �peiroi to pl joc r ∈ Q pou ikanopoioÔnthn x < r < y.

B. Basikèc ask seic

1.8 Askhseic · 27

1. DeÐxte ìti ta parak�tw isqÔoun sto R:(a) An x < y + ε gia k�je ε > 0, tìte x ≤ y.(b) An x ≤ y + ε gia k�je ε > 0, tìte x ≤ y.(g) An |x− y| ≤ ε gia k�je ε > 0, tìte x = y.(d) An a < x < b kai a < y < b, tìte |x− y| < b− a.

2. (a) An |a− b| < ε, tìte up�rqei x ¸ste

|a− x| < ε

2kai |b− x| < ε

2.

(b) IsqÔei to antÐstrofo?(g) 'Estw ìti a < b < a + ε. BreÐte ìlouc touc x ∈ R pou ikanopoioÔn tic|a− x| < ε

2 kai |b− x| < ε2 .

3. Na deiqjeÐ me epagwg ìti o arijmìc 7n−4n eÐnai pollapl�sio tou 3 gia k�jen ∈ N.4. Exet�ste gia poiec timèc tou fusikoÔ arijmoÔ n isqÔoun oi parak�tw anisì-thtec:(i) 2n > n3, (ii) 2n > n2, (iii) 2n > n, (iv) n! > 2n, (v) 2n−1 ≤ n2.

5. 'Estw a, b ∈ R kai n ∈ N. DeÐxte ìti

an − bn = (a− b)n−1∑

k=0

akbn−1−k.

An 0 < a < b, deÐxte ìti

nan−1 ≤ bn − an

b− a≤ nbn−1.

6. 'Estw a ∈ R. DeÐxte ìti:(a) An a > 1, tìte an > a gia k�je fusikì arijmì n ≥ 2.(b) An a > 1 kai m,n ∈ N, tìte am < an an kai mìno an m < n.(g) An 0 < a < 1, tìte an < a gia k�je fusikì arijmì n ≥ 2.(d) An 0 < a < 1 kai m,n ∈ N, tìte am < an an kai mìno an m > n.

7. 'Estw a ∈ R kai èstw n ∈ N. DeÐxte ìti:(a) An a ≥ −1, tìte (1 + a)n ≥ 1 + na.(b) An 0 < a < 1/n, tìte (1 + a)n < 1/(1− na).(g) An 0 ≤ a ≤ 1, tìte

1− na ≤ (1− a)n ≤ 11 + na

.

8. 'Estw a ∈ R. DeÐxte ìti:(a) An −1 < a < 0, tìte (1 + a)n ≤ 1 + na + n(n−1)

2 a2 gia k�je n ∈ N.

(b) An a > 0, tìte (1 + a)n ≥ 1 + na + n(n−1)2 a2 gia k�je n ∈ N.

9. DeÐxte ìti gia k�je n ∈ N isqÔoun oi anisìthtec(

1 +1n

)n

<

(1 +

1n + 1

)n+1

kai(

1 +1n

)n+1

>

(1 +

1n + 1

)n+2

.

10. (Anisìthta arijmhtikoÔ-gewmetrikoÔ mèsou) An x1, . . . , xn > 0, tìte

x1x2 · · ·xn ≤(

x1 + · · ·+ xn

n

)n

.

Isìthta isqÔei an kai mìno an x1 = x2 = · · · = xn.

EpÐshc, an x1, x2, . . . , xn > 0, tìte

x1x2 · · ·xn ≥(

n1x1

+ · · ·+ 1xn

)n

.

11. DeÐxte ìti k�je mh kenì k�tw fragmèno uposÔnolo A tou R èqei mègistok�tw fr�gma.

12. 'Estw A mh kenì uposÔnolo tou R kai èstw a0 ∈ A me thn idiìthta: giak�je a ∈ A, a ≤ a0. DeÐxte ìti a0 = sup A.

13. 'Estw A mh kenì fragmèno uposÔnolo tou R me inf A = sup A. Ti sumpe-raÐnete gia to A?

14. Na brejoÔn, an up�rqoun, ta max, min, sup kai inf twn parak�tw sunìlwn:

(a) A = {x > 0 : 0 < x2 − 1 ≤ 2}, B = {x ∈ Q : x ≥ 0, 0 < x2 − 1 ≤ 2}.(b) Γ = {x ∈ R : x ≥ 0, 0 < 1/(x2 − 1) ≤ 1}, ∆ = { 1

n : n ∈ N}.(g) E = { 1

n : n ∈ Z, n 6= 0}, Z = {0, 12 , 1

3 , 14}.

(d) H = {0, 12 , 1

3 , 14 , . . .}, Θ = {x ∈ R : x2 + x + 1 ≥ 0}.

(e) I = {x ∈ R : x2 + x + 1 < 0}, K = {x ∈ R : x < 0, x2 + x− 1 < 0}.(st) Λ = { 1

n + (−1)n : n ∈ N}, M = {x ∈ Q : (x− 1)(x +√

2) < 0}.(z) N = {5 + 6

n : n ∈ N} ∪ {7− 8n : n ∈ N}.15. BreÐte to supremum kai to infimum twn sunìlwn

A ={

1 + (−1)n +(−1)n+1

n: n ∈ N

}, B =

{12n

+1

3m: n,m ∈ N

}.

16. DeÐxte ìti to sÔnolo

A ={

(−1)nm

n + m: m,n = 1, 2, . . .

}

eÐnai fragmèno kai breÐte ta supA kai inf A. Exet�ste an to A èqei mègisto el�qisto stoiqeÐo.

1.8 Askhseic · 29

17. 'Estw A,B mh ken� fragmèna uposÔnola tou R me A ⊆ B. DeÐxte ìti

inf B ≤ inf A ≤ sup A ≤ supB.

18. 'Estw A,B mh ken�, fragmèna uposÔnola tou R. DeÐxte ìti to A ∪B eÐnaifragmèno kai

sup(A ∪B) = max{supA, supB} , inf(A ∪B) = min{inf A, inf B}.

MporoÔme na poÔme k�ti an�logo gia to sup(A ∩B) to inf(A ∩B)?

19. 'Estw A,B mh ken� uposÔnola tou R. DeÐxte ìti sup A ≤ inf B an kai mìnoan gia k�je a ∈ A kai gia k�je b ∈ B isqÔei a ≤ b.

20. 'Estw A,B mh ken� uposÔnola tou R pou ikanopoioÔn ta ex c:(a) gia k�je a ∈ A kai gia k�je b ∈ B isqÔei a < b.(b) A ∪B = R.

DeÐxte ìti up�rqei γ ∈ R tètoioc ¸ste eÐte A = (−∞, γ) kai B = [γ, +∞) A = (−∞, γ] kai B = (γ, +∞).

21. 'Estw A,B mh ken�, �nw fragmèna uposÔnola tou R me thn ex c idiìthta:gia k�je a ∈ A up�rqei b ∈ B ¸ste

a ≤ b.

DeÐxte ìti sup A ≤ supB.

22. 'Estw a, b ∈ R me a < b. BreÐte to supremum kai to infimum tou sunìlou(a, b) ∩Q = {x ∈ Q : a < x < b}. Aitiolog ste pl rwc thn ap�nthsh sac.

23. Gia k�je x ∈ R orÐzoume Ax = {q ∈ Q : q < x}. DeÐxte ìti

x = y ⇐⇒ Ax = Ay.

G. Ask seic

1. (a) Anisìthta Cauchy-Schwarz:(

n∑

k=1

akbk

)2

≤(

n∑

k=1

a2k

) (n∑

k=1

b2k

).

(b) Anisìthta tou Minkowski:(

n∑

k=1

(ak + bk)2)1/2

≤(

n∑

k=1

a2k

)1/2

+

(n∑

k=1

b2k

)1/2

.

2. (Tautìthta tou Lagrange) An a1, . . . , an ∈ R kai b1, . . . , bn ∈ R, tìte(

n∑

k=1

a2k

)(n∑

k=1

b2k

)−

(n∑

k=1

akbk

)2

=12

n∑

k,j=1

(akbj − ajbk)2.

Qrhsimopoi¸ntac thn tautìthta tou Lagrange deÐxte thn anisìthta Cauchy-Schwarz.

3. 'Estw a1, . . . , an > 0. DeÐxte ìti

(a1 + a2 + · · ·+ an)(

1a1

+1a2

+ · · ·+ 1an

)≥ n2.

4. An a > 0, b > 0 kai a + b = 1, tìte

2

[(a +

1a

)2

+(

b +1b

)2]≥ 25.

5. (a) An a1, . . . , an > 0, deÐxte ìti

(1 + a1) · · · (1 + an) ≥ 1 + a1 + · · ·+ an.

(b) An 0 < a1, . . . , an < 1, tìte

1− (a1 + · · ·+ an) ≤ (1− a1) · · · (1− an)

≤ 1− (a1 + · · ·+ an) + (a1a2 + a1a3 + · · ·+ an−1an).

6*. An a1 ≥ a2 ≥ · · · ≥ an > 0 kai b1 ≥ b2 ≥ · · · ≥ bn > 0, tìte

a1bn + a2bn−1 + · · ·+ anb1

n≤ a1 + · · ·+ an

n· b1 + · · ·+ bn

n

≤ a1b1 + · · ·+ anbn

n.

7*. 'Estw a1, . . . , an jetikoÐ pragmatikoÐ arijmoÐ. DeÐxte ìti up�rqei 1 ≤ m ≤n− 1 me thn idiìthta

∣∣∣∣m∑

k=1

ak −n∑

k=m+1

ak

∣∣∣∣ ≤ max{a1, . . . , an}.

Upìdeixh: Jewr ste touc arijmoÔc

bm =m∑

k=1

ak −n∑

k=m+1

ak, m = 1, . . . , n− 1

kai

b0 = −n∑

k=1

ak , bn =n∑

k=1

ak.

DeÐxte ìti dÔo diadoqikoÐ apì autoÔc eÐnai eterìshmoi.

8. 'Estw A,B mh ken�, fragmèna uposÔnola tou R. OrÐzoume A + B = {a + b :a ∈ A, b ∈ B}. DeÐxte ìti

sup(A + B) = sup A + sup B , inf(A + B) = inf A + inf B.

1.8 Askhseic · 31

9. 'Estw A,B mh ken�, fragmèna sÔnola jetik¸n pragmatik¸n arijm¸n. OrÐ-zoume A ·B = {ab : a ∈ A, b ∈ B}. DeÐxte ìti

sup(A ·B) = sup A · supB , inf(A ·B) = inf A · inf B.

10. 'Estw A mh kenì, fragmèno uposÔnolo tou R. An t ∈ R, orÐzoume tA ={ta : a ∈ A}. DeÐxte ìti

(a) an t ≥ 0 tìte sup(tA) = t supA kai inf(tA) = t inf A.(b) an t < 0 tìte sup(tA) = t inf A kai inf(tA) = t supA.

Kef�laio 2

AkoloujÐec pragmatik¸narijm¸n

2.1 AkoloujÐec pragmatik¸n arijm¸n

Orismìc 2.1.1. AkoloujÐa lègetai k�je sun�rthsh a : N → R (me pedÐoorismoÔ to sÔnolo twn fusik¸n arijm¸n kai timèc stouc pragmatikoÔc arijmoÔc).AntÐ na sumbolÐzoume tic timèc thc akoloujÐac a me a(1), a(2), . . ., gr�foume

a1, a2, a3, . . .

kai lème ìti o arijmìc an eÐnai o n-ostìc ìroc thc akoloujÐac. H Ðdia h akolou-jÐa sumbolÐzetai me {an}∞n=1, {an}, (an), (a1, a2, a3, . . .) qwrÐc autì na prokaleÐsÔgqush.

ParadeÐgmata 2.1.2. (a) 'Estw c ∈ R. H akoloujÐa an = c, n = 1, 2, . . .

lègetai stajer akoloujÐa me tim c.

(b) an = n. Oi pr¸toi ìroi thc (an) eÐnai: a1 = 1, a2 = 2, a3 = 3.

(g) an = 1n . Oi pr¸toi ìroi thc (an) eÐnai: a1 = 1, a2 = 1

2 , a3 = 13 .

(d) an = an, ìpou a ∈ R. Oi pr¸toi ìroi thc (an) eÐnai: a1 = a, a2 = a2,a3 = a3.

(e) a1 = 1 kai an+1 =√

1 + an, n = 1, 2, . . .. Aut h akoloujÐa orÐzetai ana-dromik�: an gnwrÐzoume ton an tìte mporoÔme na upologÐsoume ton an+1 qrh-simopoi¸ntac thn an+1 =

√1 + an. Dedomènou ìti èqei dojeÐ o pr¸toc thc

ìroc, h (an) eÐnai kal� orismènh (k�nontac n − 1 b mata mporoÔme na broÔmeton an). Oi pr¸toi ìroi thc (an) eÐnai: a1 = 1, a2 =

√2, a3 =

√1 +

√2,

a4 =√

1 +√

1 +√

2.

(st) a1 = 1, a2 = 1 kai an+2 = an + an+1, n = 1, 2, . . .. An gnwrÐzoumetouc an kai an+1 tìte mporoÔme na upologÐsoume ton an+2 qrhsimopoi¸ntac thnanadromik sqèsh an+2 = an + an+1. Dedomènou ìti èqoun dojeÐ oi pr¸toi dÔo

34 · Akoloujiec pragmatikwn arijmwn

ìroi, h (an) eÐnai kal� orismènh (k�nontac n− 2 b mata mporoÔme na broÔme tonan). Oi pr¸toi ìroi thc (an) eÐnai: a1 = 1, a2 = 1, a3 = 2, a4 = 3, a5 = 5,a6 = 8.(z) an = 1

n an n = 2k kai an = 12 an n = 2k − 1. Gia ton upologismì tou n�

ostoÔ ìrou an arkeÐ na gnwrÐzoume an o n eÐnai �rtioc perittìc: gia par�deigma,a6 = 1

6 kai a7 = 12 .

Orismìc 2.1.3. 'Estw (an) kai (bn) dÔo akoloujÐec pragmatik¸n arijm¸n.(a) Lème ìti (an) = (bn) (oi akoloujÐec eÐnai Ðsec) an an = bn gia k�je n ∈ N.Dhlad ,

a1 = b1, a2 = b2, a3 = b3, . . . .

(b) To �jroisma, h diafor�, to ginìmeno kai to phlÐko twn akolouji¸n (an), (bn)eÐnai oi akoloujÐec (an + bn), (an − bn), (anbn) kai (an/bn) antÐstoiqa (gia thnteleutaÐa prèpei na k�noume thn epiplèon upìjesh ìti bn 6= 0 gia k�je n ∈ N).Orismìc 2.1.4 (sÔnolo twn ìrwn). To sÔnolo twn ìrwn thc akoloujÐac(an) eÐnai to

A = {an : n ∈ N}.Den ja prèpei na sugqèei kaneÐc thn akoloujÐa (an) = (a1, a2, . . .) me to sÔno-lo twn tim¸n thc. Gia par�deigma, to sÔnolo tim¸n thc akoloujÐac (−1)n =(1,−1, 1,−1, . . .) eÐnai to disÔnolo {−1, 1}. Parathr ste epÐshc ìti dÔo diafo-retikèc akoloujÐec mporeÐ na èqoun to Ðdio sÔnolo tim¸n (d¸ste paradeÐgmata).

Orismìc 2.1.5 (telikì tm ma). 'Estw (an) mia akoloujÐa pragmatik¸narijm¸n. K�je akoloujÐa thc morf c (am+n−1)∞n=1 = (am, am+1, am+2, . . .)ìpou m ∈ N lègetai telikì tm ma thc (an). Gia par�deigma, oi akoloujÐec(5, 6, 7, . . .) kai (30, 31, 32, . . .) eÐnai telik� tm mata thc an = n.

'Askhsh 2.1.6. 'Estw (an) mia akoloujÐa pragmatik¸n arijm¸n kai èstw(am+n−1)∞n=1 èna telikì tm ma thc. DeÐxte ìti:

(a) k�je telikì tm ma thc (am+n−1) eÐnai telikì tm ma thc (an).(b) k�je telikì tm ma thc (an) perièqei k�poio telikì tm ma thc (am+n−1).

2.2 SÔgklish akolouji¸n

2.2aþ Orismìc tou orÐou

JewroÔme tic akoloujÐec (an) kai (bn) me n�ostoÔc ìrouc touc

an =1n

kai bn = (−1)n.

Gia {meg�lec} timèc tou n oi ìroi 1/n thc (an) brÐskontai (ìlo kai pio) {kont�}sto 0. Apì thn �llh pleur�, oi ìroi (−1)n thc (bn) den plhsi�zoun se k�poionpragmatikì arijmì. Ja lègame ìti h akoloujÐa (an) sugklÐnei (èqei ìrio to 0kaj¸c to n teÐnei sto �peiro) en¸ h (bn) den sugklÐnei. Me �lla lìgia, jèloumena ekfr�soume austhr� thn prìtash:

2.2 Sugklish akoloujiwn · 35

{h (an) sugklÐnei ston a an gia meg�lec timèc tou n o an eÐnai kont�ston a}.

Autì pou prèpei na k�noume safèc eÐnai to nìhma twn fr�sewn {kont�} kai{meg�lec timèc}. Gia par�deigma, an k�poioc jewreÐ ìti h apìstash 1 eÐnai ika-nopoihtik� mikr , tìte h (an) èqei ìlouc touc ìrouc thc kont� ston 1/2. EpÐshc,an k�poioc jewreÐ ìti h fr�sh {meg�lec timèc} shmaÐnei {arketèc meg�lec timèc},tìte h (bn) èqei arketoÔc ìrouc kont� ston 1 all� kai arketoÔc ìrouc kont�ston −1. SumfwnoÔme na lème ìti:

{h (an) sugklÐnei ston a an se osod pote mikr perioq tou a brÐ-skontai telik� ìloi oi ìroi thc (an)}.

H ènnoia thc perioq c enìc pragmatikoÔ arijmoÔ a orÐzetai austhr� wc ex c: giak�je ε > 0 to anoiktì di�sthma (a−ε, a+ε) me kèntro ton a kai aktÐna ε eÐnai miaperioq tou a (h ε�perioq tou a). Qrhsimopoi¸ntac thn ènnoia thc ε�perioq ckai thn ènnoia tou telikoÔ tm matoc miac akoloujÐac, katal goume sto ex c:

{h (an) sugklÐnei ston a an k�je ε�perioq tou a perièqei k�poiotelikì tm ma thc (an)}.

Parathr¸ntac ìti x ∈ (a − ε, a + ε) an kai mìno an |x − a| < ε, mporoÔme nad¸soume ton ex c austhrì orismì.

Orismìc 2.2.1 (ìrio akoloujÐac). 'Estw (an) mia akoloujÐa pragmati-k¸n arijm¸n. Lème ìti h (an) sugklÐnei ston pragmatikì arijmì a an isqÔeito ex c:

Gia k�je ε > 0 up�rqei fusikìc n0 = n0(ε) me thn idiìthta: an n ∈ Nkai n ≥ n0(ε), tìte |an − a| < ε.

An h (an) sugklÐnei ston a, gr�foume lim an = a limn→∞

an = a , pio apl�,an → a.

Parat rhsh 2.2.2. Ston parap�nw orismì, o deÐkthc n0 exart�tai k�je for�apì to ε. 'Oso ìmwc mikrì ki an eÐnai to ε, mporoÔme na broÔme n0(ε) ¸ste ìloioi ìroi an pou èpontai tou an0 na brÐskontai{ε-kont�} ston a. SkefteÐte thnprosp�jeia epilog c tou n0(ε) san èna ep� �peiron paiqnÐdi me ènan antÐpalo oopoÐoc epilègei oloèna kai mikrìtero ε > 0.

Gia na exoikeiwjoÔme me ton orismì ja apodeÐxoume ìti h an = 1n → 0 en¸ h

bn = (−1)n den sugklÐnei (se kanènan pragmatikì arijmì).

(a) H an = 1n sugklÐnei sto 0: JewroÔme tuqoÔsa ε�perioq (−ε, ε) tou 0. Apì

thn Arqim deia idiìthta up�rqei n0(ε) ∈ N ¸ste 1n0

< ε. O mikrìteroc tètoiocfusikìc arijmìc eÐnai o

[1ε

]+ 1 (exhg ste giatÐ), ìmwc autì den èqei idiaÐterh

shmasÐa. Tìte, gia k�je n ≥ n0 isqÔei

−ε < 0 <1n≤ 1

n0< ε.

Dhlad , to telikì tm ma(

1n0

, 1n0+1 , 1

n0+2 , . . .)

thc (an) perièqetai sto (−ε, ε).Sumfwna me ton orismì, èqoume an → 0.

(b) H bn = (−1)n den sugklÐnei: Ac upojèsoume ìti up�rqei a ∈ R ¸ste (−1)n →a. DiakrÐnoume dÔo peript¸seic:

(b1) An a 6= 1 up�rqei ε�perioq tou a ¸ste 1 /∈ (a − ε, a + ε). Gia par�-deigma, mporoÔme na epilèxoume ε = |1−a|

2 . AfoÔ bn → a, up�rqei telikì tm ma(bm, bm+1, . . .) pou perièqetai sto (a − ε, a + ε). Eidikìtera, bn 6= 1 gia k�jen ≥ m. Autì eÐnai �topo: an jewr soume �rtio n ≥ m tìte bn = (−1)n = 1.

(b2) An a 6= −1 up�rqei ε�perioq tou a ¸ste −1 /∈ (a − ε, a + ε). Giapar�deigma, mporoÔme na epilèxoume ε = |1+a|

2 . AfoÔ bn → a, up�rqei telikìtm ma (bm, bm+1, . . .) pou perièqetai sto (a − ε, a + ε). Eidikìtera, bn 6= −1gia k�je n ≥ m. Autì eÐnai �topo: an jewr soume perittì n ≥ m tìte bn =(−1)n = −1.

Je¸rhma 2.2.3 (monadikìthta tou orÐou). An an → a kai an → b,tìte a = b.

Apìdeixh. Upojètoume ìti a 6= b. QwrÐc periorismì thc genikìthtac, mporoÔmena upojèsoume ìti a < b. An p�roume ε = (b−a)/4, tìte a+ ε < b− ε. Dhlad ,

(a− ε, a + ε) ∩ (b− ε, b + ε) = ∅.

AfoÔ an → a, mporoÔme na broÔme n1 ∈ N ¸ste gia k�je n ≥ n1 na isqÔei|an − a| < ε. OmoÐwc, afoÔ an → b, mporoÔme na broÔme n2 ∈ N ¸ste gia k�jen ≥ n2 na isqÔei |an − b| < ε.

Jètoume n0 = max{n1, n2}. Tìte, gia k�je n ≥ n0 isqÔoun tautìqrona oi

|an − a| < ε kai |an − b| < ε.

'Omwc tìte, gia k�je n ≥ n0 èqoume

an ∈ (a− ε, a + ε) ∩ (b− ε, b + ε),

to opoÐo eÐnai �topo. 2

Je¸rhma 2.2.4 (krit rio parembol c krit rio isosugklinou-s¸n akolouji¸n). JewroÔme treÐc akoloujÐec an, bn, γn pou ikanopoioÔn taex c:

(a) an ≤ bn ≤ γn gia k�je n ∈ N.(b) lim an = lim γn = `.

Tìte, h (bn) sugklÐnei kai lim bn = `.

Apìdeixh. 'Estw ε > 0. AfoÔ an → ` kai γn → `, up�rqoun fusikoÐ arijmoÐn1, n2 ¸ste

|an − `| < ε an n ≥ n1 kai |γn − `| < ε an n ≥ n2.

2.2 Sugklish akoloujiwn · 37

IsodÔnama,

`− ε < an < ` + ε an n ≥ n1 kai `− ε < γn < ` + ε an n ≥ n2.

Epilègoume n0 = max{n1, n2}. An n ≥ n0, tìte

`− ε < an ≤ bn ≤ γn < ` + ε

dhlad , an n ≥ n0 èqoume |bn − `| < ε. Me b�sh ton orismì, bn → `. 2

Parathr seic 2.2.5. (a) BebaiwjeÐte ìti èqete katal�bei th diadikasÐa a-pìdeixhc: an jèloume na deÐxoume ìti tn → t, prèpei gia aujaÐreto (mikrì) ε > 0� h apìdeixh xekin�ei me thn fr�sh {èstw ε > 0} � na broÔme fusikì n0 (pouexart�tai apì to ε) me thn idiìthta: n ≥ n0(ε) =⇒ |tn − t| < ε.(b) 'Iswc èqete dh parathr sei ìti oi pr¸toi m ìroi (m = 2, 10 kai 1010) denephre�zoun th sÔgklish mh miac akoloujÐac. Qrhsimopoi¸ntac thn 'Askhsh2.1.6 deÐxte ta ex c:1. 'Estw m ∈ N. H akoloujÐa (an) sugklÐnei an kai mìno an h akoloujÐa(bn) = (am+n−1) sugklÐnei, kai m�lista limn an = limn am+n−1.2. 'Estw (an) kai (bn) dÔo akoloujÐec pou diafèroun se peperasmènouc to pl jocìrouc: up�rqei m ∈ N ¸ste an = bn gia k�je n ≥ m. An h (an) sugklÐnei stona tìte h (bn) sugklÐnei ki aut ston a.

Orismìc 2.2.6. H akoloujÐa (an) lègetai fragmènh an mporoÔme na broÔmek�poion M > 0 me thn idiìthta

|an| ≤ M gia k�je n ∈ N.

Je¸rhma 2.2.7. K�je sugklÐnousa akoloujÐa eÐnai fragmènh.

Apìdeixh. 'Estw ìti an → a ∈ R. PaÐrnoume ε = 1 > 0. MporoÔme na broÔmen0 ∈ N ¸ste |an − a| < 1 gia k�je n ≥ n0. Dhlad ,

an n ≥ n0, tìte |an| ≤ |an − a|+ |a| < 1 + |a|.Jètoume

M = max{|a1|, . . . , |an0 |, 1 + |a|}kai eÔkola elègqoume ìti |an| ≤ M gia k�je n ∈ N (diakrÐnete peript¸seic:n ≤ n0 kai n > n0). 'Ara, h (an) eÐnai fragmènh. 2

2.2bþ AkoloujÐec pou teÐnoun sto �peiro

Orismìc 2.2.8. 'Estw (an) mia akoloujÐa pragmatik¸n arijm¸n.(a) Lème ìti an → +∞ (h akoloujÐa teÐnei sto +∞) an gia k�je M > 0(osod pote meg�lo) up�rqei fusikìc n0 = n0(M) ¸ste

an n ≥ n0, tìte an > M.

(b) Lème ìti an → −∞ (h akoloujÐa teÐnei sto −∞) an gia k�je M > 0(osod pote meg�lo) up�rqei fusikìc n0 = n0(M) ¸ste

an n ≥ n0, tìte an < −M.

Parat rhsh 2.2.9. Qrhsimopoi same th lèxh {teÐnei} sto ±∞: sumfwnoÔmepwc mia akoloujÐa (an) sugklÐnei mìno an sugklÐnei se k�poion pragmatikìarijmì a (o opoÐoc lègetai kai ìrio thc (an)). Se ìlec tic �llec peript¸seic jalème ìti h akoloujÐa apoklÐnei.

2.2gþ H �rnhsh tou orismoÔ

KleÐnoume aut n thn Par�grafo me thn akrib diatÔpwsh thc �rnhshc tou ori-smoÔ tou orÐou. JumhjeÐte ìti:

{h (an) sugklÐnei ston a an k�je ε�perioq tou a perièqei k�poiotelikì tm ma thc (an)}.

Epomènwc, h (an) den sugklÐnei ston a an up�rqei perioq (a − ε, a + ε)tou a h opoÐa den perièqei kanèna telikì tm ma thc (an). IsodÔnama,

{h (an) den sugklÐnei ston a an up�rqei ε > 0 ¸ste: k�je telikìtm ma (am, am+1, . . .) thc (an) èqei toul�qiston ènan ìro pou denan kei sto (a− ε, a + ε)}.

Parathr ste ìti an (am, am+1, . . .) eÐnai èna telikì tm ma thc (an) tìte: to(am, am+1, . . .) den perièqetai sto (a − ε, a + ε) an kai mìno an up�rqei n ≥ m

¸ste an /∈ (a − ε, a + ε), dhlad |an − a| ≥ ε. Katal goume loipìn sthn ex cprìtash:

{h (an) den sugklÐnei ston a an up�rqei ε > 0 ¸ste: gia k�je m ∈ Nup�rqei n ≥ m ¸ste |an − a| ≥ ε}.

'Askhsh 2.2.10. DeÐxte ìti h akoloujÐa (an) den sugklÐnei ston a an kaimìno an up�rqei ε > 0 ¸ste �peiroi to pl joc ìroi thc (an) ikanopoioÔn thn|an − a| ≥ ε.

2.3 'Algebra twn orÐwn

'Olec oi basikèc idiìthtec twn orÐwn akolouji¸n apodeiknÔontai eÔkola me b�shton orismì.

Prìtash 2.3.1. an → a an kai mìno an an−a → 0 an kai mìno an |an−a| → 0.

Apìdeixh. ArkeÐ na gr�youme touc treÐc orismoÔc:

(i) 'Eqoume an → a an gia k�je ε > 0 up�rqei n0 ∈ N ¸ste gia k�je n ≥ n0

na isqÔei |an − a| < ε.

(ii) 'Eqoume an − a → 0 an gia k�je ε > 0 up�rqei n0 ∈ N ¸ste gia k�jen ≥ n0 na isqÔei |(an − a)− 0| < ε.

(iii) 'Eqoume |an − a| → 0 an gia k�je ε > 0 up�rqei n0 ∈ N ¸ste gia k�jen ≥ n0 na isqÔei

∣∣ |an − a| − 0∣∣ < ε.

2.3 Algebra twn oriwn · 39

Parathr¸ntac ìti |an − a| = |(an − a) − 0| =∣∣ |an − a| − 0

∣∣ gia k�je n ∈ Nblèpoume ìti oi treic prot�seic lène akrib¸c to Ðdio pr�gma. 2

Prìtash 2.3.2. an → 0 an kai mìno an |an| → 0.

Apìdeixh. Eidik perÐptwsh thc Prìtashc 2.3.1 (a = 0). 2

Prìtash 2.3.3. An an → a tìte |an| → |a|.Apìdeixh. 'Estw ε > 0. AfoÔ an → a, up�rqei n0 ∈ N ¸ste gia k�je n ≥ n0 naisqÔei |an − a| < ε. Tìte, gia k�je n ≥ n0 èqoume

∣∣ |an| − |a|∣∣ ≤ |an − a| < ε,

apì thn trigwnik anisìthta gia thn apìluth tim . 2

Prìtash 2.3.4. An an → a kai bn → b tìte an + bn → a + b.

Apìdeixh. 'Estw ε > 0. AfoÔ an → a, up�rqei n1 ∈ N ¸ste gia k�je n ≥ n1 naisqÔei

|an − a| < ε

2.

OmoÐwc, afoÔ bn → b, up�rqei n2 ∈ N ¸ste gia k�je n ≥ n2 na isqÔei

|bn − b| < ε

2.

Jètoume n0 = max{n1, n2}. Tìte, gia k�je n ≥ n0 èqoume tautìqrona |an−a| <ε/2 kai |bn − b| < ε/2. 'Ara, gia k�je n ≥ n0 èqoume

|(an + bn)− (a + b)| = |(an − a) + (bn − b)| ≤ |an − a|+ |bn − b| < ε

2+

ε

2= ε.

AfoÔ to ε > 0 tan tuqìn, autì deÐqnei ìti an + bn → a + b. 2

Prìtash 2.3.5. 'Estw (an) kai (bn) dÔo akoloujÐec. Upojètoume ìti h (bn)eÐnai fragmènh kai ìti an → 0. Tìte, anbn → 0.

Apìdeixh. H (bn) eÐnai fragmènh, �ra up�rqei M > 0 ¸ste |bn| ≤ M gia k�jen ∈ N. 'Estw ε > 0. AfoÔ an → 0, up�rqei n0 ∈ N ¸ste

|an| = |an − 0| < ε

M

gia k�je n ≥ n0. 'Epetai ìti, an n ≥ n0 tìte

|anbn| = |an||bn| < ε

M·M = ε.

AfoÔ to ε > 0 tan tuqìn, autì deÐqnei ìti anbn → 0. 2

Prìtash 2.3.6. An an → a kai t ∈ R tìte tan → ta.

Apìdeixh. Apì thn an → a èpetai ìti an − a → 0. JewroÔme thn stajer akoloujÐa bn = t. Apì thn prohgoÔmenh Prìtash èqoume

tan − ta = t(an − a) = bn(an − a) → 0.

Sunep¸c, tan → ta. 2

Prìtash 2.3.7. An an → a kai bn → b, tìte anbn → ab.

Apìdeixh. Gr�foume

anbn − ab = an(bn − b) + b(an − a).

ParathroÔme ta ex c:

(i) H (an) sugklÐnei, �ra eÐnai fragmènh. AfoÔ bn − b → 0, h Prìtash 2.3.5deÐqnei ìti an(bn − b) → 0.

(ii) AfoÔ an − a → 0, h Prìtash 2.3.6 deÐqnei ìti b(an − a) → 0.

T¸ra, h Prìtash 2.3.4 deÐqnei ìti

an(bn − b) + b(an − a) → 0 + 0 = 0.

Dhlad , anbn − ab → 0. 2

Prìtash 2.3.8. 'Estw k ∈ N, k ≥ 2. An an → a tìte akn → ak.

Apìdeixh. Me epagwg wc proc k. An an → a kai an gnwrÐzoume ìti amn → am,

tìteam+1

n = an · amn → a · am = am+1

apì thn Prìtash 2.3.7. 2

Prìtash 2.3.9. 'Estw (an) kai (bn) akoloujÐec me bn 6= 0 gia k�je n ∈ N.An an → a kai bn → b 6= 0, tìte an

bn→ a

b .

Apìdeixh. ArkeÐ na deÐxoume ìti 1bn→ 1

b . Katìpin, efarmìzoume thn Prìtash

2.3.7 gia tic (an) kai(

1bn

).

Autì pou jèloume na gÐnei mikrì gia meg�lec timèc tou n eÐnai h posìthta∣∣∣∣

1bn− 1

b

∣∣∣∣ =|b− bn||bn||b| .

Isqurismìc. Up�rqei n1 ∈ N ¸ste: gia k�je n ≥ n1,

|bn| > |b|2

.

Gia thn apìdeixh autoÔ tou isqurismoÔ epilègoume ε = |b|2 > 0 kai, lìgw thc

bn → b, brÐskoume n1 ∈ N ¸ste: an n ≥ n1 tìte |bn − b| < |b|2 . Tìte, gia k�je

n ≥ n1 isqÔei∣∣ |bn| − |b|

∣∣ ≤ |bn − b| < |b|2

.

Apì thn teleutaÐa anisìthta èpetai ìti |bn| > |b|2 gia k�je n ≥ n1.

O isqurismìc èqei thn ex c sunèpeia: an n ≥ n1 tìte∣∣∣∣

1bn− 1

b

∣∣∣∣ ≤2|b− bn||b|2 .

2.3 Algebra twn oriwn · 41

T¸ra mporoÔme na deÐxoume ìti 1bn→ 1

b . 'Estw ε > 0. AfoÔ bn → b, up�rqei

n2 ∈ N ¸ste |b − bn| < ε|b|22 gia k�je n ≥ n2. Epilègoume n0 = max{n1, n2}.

An n ≥ n0, tìte ∣∣∣∣1bn− 1

b

∣∣∣∣ ≤2|b− bn||b|2 < ε.

Me b�sh ton orismì, 1bn→ 1

b . 2

Prìtash 2.3.10. 'Estw k ∈ N, k ≥ 2. An an ≥ 0 gia k�je n ∈ N kai anan → a, tìte k

√an → k

√a.

Apìdeixh. DiakrÐnoume dÔo peript¸seic:

(a) an → 0: 'Estw ε > 0. AfoÔ an → 0, efarmìzontac ton orismì gia ton jetikìarijmì ε1 = εk brÐskoume n0 ∈ N ¸ste: gia k�je n ≥ n0 isqÔei

0 ≤ an < εk.

Tìte, gia k�je n ≥ n0 isqÔei

0 ≤ k√

an <k√

εk = ε.

'Ara, k√

an → 0.

(b) an → a > 0: JumhjeÐte ìti an x, y ≥ 0 tìte

|xk − yk| = |x− y|(xk−1 + xk−2y + · · ·+ xyk−2 + yk−1) ≥ |x− y|yk−1.

Qrhsimopoi¸ntac aut n thn anisìthta me x = k√

an kai y = k√

a blèpoume ìti

∣∣ k√

an − k√

a∣∣ ≤ |an − a|

k√

ak−1.

'Estw ε > 0. AfoÔ an → 0, efarmìzontac ton orismì gia ton jetikì arijmìε1 = k

√ak−1 · ε, brÐskoume n0 ∈ N ¸ste: gia k�je n ≥ n0 isqÔei

|an − a| < k√

ak−1 · ε.

Tìte, gia k�je n ≥ n0 isqÔei

∣∣ k√

an − k√

a∣∣ ≤ |an − a|

k√

ak−1< ε.

Sunep¸c, k√

an → k√

a. 2

Prìtash 2.3.11. An an ≤ bn gia k�je n ∈ N kai an an → a, bn → b, tìtea ≤ b.

Apìdeixh. Upojètoume ìti a > b. An jèsoume ε = a−b2 tìte up�rqoun n1, n2 ∈ N

¸ste: gia k�je n ≥ n1 isqÔei

|an − a| < a− b

2=⇒ an > a− a− b

2=

a + b

2,

kai gia k�je n ≥ n2 isqÔei

|bn − b| < a− b

2=⇒ bn < b +

a− b

2=

a + b

2.

Jètoume n0 = max{n1, n2}. Tìte, gia k�je n ≥ n0 èqoume

bn <a + b

2< an,

to opoÐo eÐnai �topo. 2

Prìtash 2.3.12. An m ≤ an ≤ M gia k�je n ∈ N kai an an → a, tìtem ≤ a ≤ M .

Apìdeixh. JewroÔme tic stajerèc akoloujÐec bn = m, γn = M kai efarmìzoumethn prohgoÔmenh Prìtash. 2

2.4 Basik� ìria kai basik� krit ria sÔgklishc

Se aut thn Par�grafo brÐskoume ta ìria k�poiwn sugkekrimènwn akolouji¸noi opoÐec emfanÐzontai polÔ suqn� sth sunèqeia. Me th bo jeia aut¸n twn {ba-sik¸n orÐwn} apodeiknÔoume dÔo polÔ qr sima krit ria sÔgklishc akolouji¸nsto 0 sto +∞.

2.4aþ Basik� ìria

Prìtash 2.4.1. An a > 1, tìte h akoloujÐa xn = an teÐnei sto +∞.

Apìdeixh. AfoÔ a > 1, up�rqei θ > 0 ¸ste a = 1 + θ. Apì thn anisìthtaBernoulli paÐrnoume

xn = (1 + θ)n ≥ 1 + nθ > nθ

gia k�je n ∈ N.'Estw t¸ra M > 0. Apì thn Arqim deia idiìthta, up�rqei n0 ∈ N ¸ste

n0 > M/θ. Tìte, gia k�je n ≥ n0 èqoume

xn > nθ ≥ n0θ > m.

'Epetai ìti xn → +∞. 2

Prìtash 2.4.2. An 0 < a < 1, tìte h akoloujÐa xn = an sugklÐnei sto 0.

Apìdeixh. 'Eqoume 1a > 1, �ra up�rqei θ > 0 ¸ste 1

a = 1+ θ. Apì thn anisìthtaBernoulli paÐrnoume

1xn

= (1 + θ)n ≥ 1 + nθ > nθ

dhlad

0 < xn <1nθ

gia k�je n ∈ N. Apì thn 1nθ → 0 kai apì to krit rio twn isosugklinous¸n

akolouji¸n èpetai ìti xn → 0. 2

2.4 Basika oria kai basika krithria sugklishc · 43

Prìtash 2.4.3. An a > 0, tìte h akoloujÐa xn = n√

a → 1.

Apìdeixh. (a) Exet�zoume pr¸ta thn perÐptwsh a > 1. Tìte, n√

a > 1 gia k�jen ∈ N. OrÐzoume

θn = n√

a− 1 = xn − 1.

Parathr ste ìti θn > 0 gia k�je n ∈ N. An deÐxoume ìti θn → 0, tìte èqoumeto zhtoÔmeno: xn = 1 + θn → 1.

AfoÔ n√

a = 1 + θn, mporoÔme na gr�youme

a = (1 + θn)n ≥ 1 + nθn > nθn.

'Epetai ìti0 < θn <

a

n,

kai apì to krit rio twn isosugklinous¸n akolouji¸n sumperaÐnoume ìti θn → 0.Sunep¸c, xn = 1 + θn → 1.(b) An 0 < a < 1 tìte 1

a > 1. Apì to (a) èqoume

1xn

=1

n√

a= n

√1a→ 1 6= 0.

Sunep¸c, xn → 1.(g) Tèloc, an a = 1 tìte xn = n

√1 = 1 gia k�je n ∈ N. EÐnai t¸ra fanerì ìti

xn → 1. 2

Prìtash 2.4.4. H akoloujÐa xn = n√

n → 1.

Apìdeixh. MimoÔmaste thn apìdeixh thc prohgoÔmenhc Prìtashc. OrÐzoume

θn = n√

n− 1 = xn − 1.

Parathr ste ìti θn > 0 gia k�je n ∈ N. An deÐxoume ìti θn → 0, tìte èqoumeto zhtoÔmeno: xn = 1 + θn → 1.

AfoÔ n√

n = 1 + θn, qrhsimopoi¸ntac to diwnumikì an�ptugma, mporoÔme nagr�youme

n = (1 + θn)n ≥ 1 + nθn +(

n

2

)θ2

n >n(n− 1)

2θ2

n.

'Epetai ìti, gia n ≥ 2,

0 < θn <

√2

n− 1,

kai apì to krit rio twn isosugklinous¸n akolouji¸n sumperaÐnoume ìti θn → 0.Sunep¸c, xn = 1 + θn → 1. 2

2.4bþ Krit rio thc rÐzac kai krit rio tou lìgou

Prìtash 2.4.5 (krit rio tou lìgou). 'Estw (an) akoloujÐa mh mhdeni-k¸n ìrwn (an 6= 0).(a) An an > 0 gia k�je n ∈ N kai an+1

an→ ` > 1, tìte an → +∞.

(b) An∣∣∣∣an+1

an

∣∣∣∣ → ` < 1, tìte an → 0.

Apìdeixh. (a) Jètoume ε = `−12 > 0. AfoÔ an+1

an→ `, up�rqei n0 ∈ N ¸ste: gia

k�je n ≥ n0,an+1

an> `− ε =

` + 12

.

Parathr ste ìti θ := `+12 > 1. Tìte, an0+1 > θan0 , an0+2 > θ2an0 , an0+3 >

θ3an0 , kai genik�, an n > n0 isqÔei (exhg ste giatÐ)

an > θn−n0an0 =an0

θn0· θn.

AfoÔ limn→∞

θn = +∞, èpetai ìti an → +∞.

(b) Jètoume ε = 1−`2 > 0. AfoÔ

∣∣∣an+1an

∣∣∣ → `, up�rqei n0 ∈ N ¸ste: gia k�jen ≥ n0, ∣∣∣∣

an+1

an

∣∣∣∣ < ` + ε =` + 1

2.

Parathr ste ìti ρ := `+12 < 1. Tìte, |an0+1| < ρ|an0 |, |an0+2| < ρ2|an0 |,

|an0+3| < ρ3|an0 |, kai genik�, an n > n0 isqÔei (exhg ste giatÐ)

|an| < ρn−n0 |an0 | =|an0 |ρn0

· ρn.

AfoÔ limn→∞

ρn = 0, èpetai ìti an → 0. 2

Parat rhsh 2.4.6. An an+1an

→ 1 tìte to krit rio den dÐnei sumpèrasma. Giapar�deigma, n+1

n → 1 kai n →∞, ìmwc 1/(n+1)1/n → 1 kai 1/n → 0.

Entel¸c an�loga apodeiknÔetai h akìloujh Prìtash.

Prìtash 2.4.7. (a) 'Estw µ > 1 kai (an) akoloujÐa jetik¸n ìrwn. An an+1 ≥µan gia k�je n, tìte an → +∞.(b) 'Estw 0 < µ < 1 kai (an) akoloujÐa me thn idiìthta |an+1| ≤ µ|an| gia k�jen. Tìte, an → 0. 2

Prìtash 2.4.8 (krit rio thc rÐzac). 'Estw (an) akoloujÐa me mh arnh-tikoÔc ìrouc.(a) An n

√an → ρ < 1 tìte an → 0.

(b) An n√

an → ρ > 1 tìte an → +∞.

Apìdeixh. (a) Jètoume ε = 1−ρ2 > 0. AfoÔ n

√an → ρ, up�rqei n0 ∈ N ¸ste: gia

k�je n ≥ n0,n√

an < ρ + ε =ρ + 1

2.

Parathr ste ìti θ := ρ+12 < 1 kai

0 ≤ an ≤ θn gia k�je n ≥ n0.

AfoÔ 0 < θ < 1, èqoume limn→∞

θn = 0. Apì to krit rio twn isosugklinous¸nakolouji¸n èpetai ìti an → 0.

2.5 Sugklish monotonwn akoloujiwn · 45

(b) Jètoume ε = ρ−12 > 0. AfoÔ n

√an → ρ, up�rqei n0 ∈ N ¸ste: gia k�je

n ≥ n0,n√

an > ρ− ε =ρ + 1

2.

Parathr ste ìti θ := ρ+12 > 1 kai

an ≥ θn gia k�je n ≥ n0.

AfoÔ θ > 1, èqoume limn→∞

θn = +∞. 'Epetai ìti an → +∞. 2

Parat rhsh 2.4.9. An n√

an → 1 tìte to krit rio den dÐnei sumpèrasma. Giapar�deigma, n

√n → 1 kai n →∞, ìmwc n

√1/n → 1 kai 1/n → 0.

Entel¸c an�loga apodeiknÔetai h ex c Prìtash.

Prìtash 2.4.10. 'Estw (an) akoloujÐa me mh arnhtikoÔc ìrouc.(a) An up�rqei 0 < ρ < 1 ¸ste n

√an ≤ ρ gia k�je n ∈ N tìte an → 0.

(b) An up�rqei ρ > 1 ¸ste n√

an ≥ ρ gia k�je n ∈ N tìte an → +∞. 2

2.5 SÔgklish monìtonwn akolouji¸n

2.5aþ SÔgklish monìtonwn akolouji¸n

Orismìc 2.5.1. 'Estw (an) mia akoloujÐa pragmatik¸n arijm¸n. Lème ìti h(an) eÐnai

(i) aÔxousa, an an+1 ≥ an gia k�je n ∈ N.(ii) fjÐnousa, an an+1 ≤ an gia k�je n ∈ N.(iii) gnhsÐwc aÔxousa, an an+1 > an gia k�je n ∈ N.(iv) gnhsÐwc fjÐnousa, an an+1 < an gia k�je n ∈ N.Se kajemÐa apì tic parap�nw peript¸seic lème ìti h (an) eÐnai monìtonh.

Parathr seic 2.5.2. (a) EÔkola elègqoume ìti an h (an) eÐnai aÔxousa tìte

n ≤ m =⇒ an ≤ am.

DeÐxte to me epagwg : stajeropoi ste to n kai deÐxte ìti an an ≤ am tìte an ≤am+1. AntÐstoiqo sumpèrasma isqÔei gia ìlouc touc �llouc tÔpouc monotonÐac.(b) K�je gnhsÐwc aÔxousa akoloujÐa eÐnai aÔxousa kai k�je gnhsÐwc fjÐnousaakoloujÐa eÐnai fjÐnousa.(g) K�je aÔxousa akoloujÐa eÐnai k�tw fragmènh, gia par�deigma apì ton pr¸tothc ìro a1. Sunep¸c, mia aÔxousa akoloujÐa eÐnai fragmènh an kai mìno an eÐnai�nw fragmènh.

Entel¸c an�loga, k�je fjÐnousa akoloujÐa eÐnai �nw fragmènh, gia par�deig-ma apì ton pr¸to thc ìro a1. Sunep¸c, mia fjÐnousa akoloujÐa eÐnai fragmènhan kai mìno an eÐnai k�tw fragmènh.

H diaÐsjhsh upodeiknÔei ìti an mia akoloujÐa eÐnai monìtonh kai fragmènh,tìte prèpei na sugklÐnei. Gia par�deigma, an h (an) eÐnai aÔxousa kai �nw frag-mènh, tìte oi ìroi thc susswreÔontai sto el�qisto �nw fr�gma tou sunìlouA = {an : n ∈ N}. Ja d¸soume austhr apìdeixh gi� autì:

Je¸rhma 2.5.3 (sÔgklish monìtonwn akolouji¸n). K�je monìtonhkai fragmènh akoloujÐa sugklÐnei.

Apìdeixh. QwrÐc periorismì thc genikìthtac upojètoume ìti h (an) eÐnai aÔxousa.To sÔnolo A = {an : n ∈ N} eÐnai mh kenì (gia par�deigma, a1 ∈ A) kai �nwfragmèno diìti h (an) eÐnai (�nw) fragmènh. Apì to axÐwma thc plhrìthtac,up�rqei to el�qisto �nw fr�gma tou. 'Estw a = sup A. Ja deÐxoume ìti an → a.

'Estw ε > 0. AfoÔ a− ε < a, o a− ε den eÐnai �nw fr�gma tou A. Dhlad ,up�rqei stoiqeÐo tou A pou eÐnai megalÔtero apì ton a − ε. Me �lla lìgia,up�rqei n0 ∈ N ¸ste

a− ε < an0 .

AfoÔ h an eÐnai aÔxousa, gia k�je n ≥ n0 èqoume an0 ≤ an kai epeid o a eÐnai�nw fr�gma tou A, an ≤ a. Dhlad , an n ≥ n0 tìte

a− ε < an0 ≤ an ≤ a < a + ε

'Epetai ìti |an − a| < ε gia k�je n ≥ n0. AfoÔ to ε > 0 tan tuqìn, sumperaÐ-noume ìti an → a. 2

Me parìmoio trìpo apodeiknÔontai ta ex c:

(i) An h (an) eÐnai fjÐnousa kai k�tw fragmènh, tìte an → inf{an : n ∈ N}.

(ii) An h (an) eÐnai aÔxousa kai den eÐnai �nw fragmènh, tìte teÐnei sto +∞.

(iii) An h (an) eÐnai fjÐnousa kai den eÐnai k�tw fragmènh, tìte teÐnei sto −∞.

Ac doÔme gia par�deigma thn apìdeixh tou deÔterou isqurismoÔ: 'Estw M > 0.AfoÔ h (an) den eÐnai �nw fragmènh, o M den eÐnai �nw fr�gma tou sunìlouA = {an : n ∈ N}. Sunep¸c, up�rqei n0 ∈ N ¸ste an0 > M . AfoÔ h (an) eÐnaiaÔxousa, gia k�je n ≥ n0 èqoume

an ≥ an0 > M.

AfoÔ o M > 0 tan tuq¸n, an → +∞. 2

2.5bþ O arijmìc e

Qrhsimopoi¸ntac to je¸rhma sÔgklishc monìtonwn akolouji¸n ja orÐsoume tonarijmì e kai ja doÔme p¸c mporeÐ kaneÐc sqetik� eÔkola na epitÔqei kalèc pro-seggÐseic tou.

Prìtash 2.5.4. H akoloujÐa an =(1 + 1

n

)n sugklÐnei se k�poion pragmatikìarijmì pou an kei sto (2, 3). OrÐzoume e := lim

n→∞(1 + 1

n

)n.

Apìdeixh. Ja deÐxoume ìti h (an) eÐnai gnhsÐwc aÔxousa kai �nw fragmènh.(a) Jèloume na elègxoume ìti an < an+1 gia k�je n ∈ N. ParathroÔme ìti

(1 +

1n

)n

<

(1 +

1n + 1

)n+1

⇐⇒(

n + 1n

)n

<

(n + 2n + 1

)nn + 2n + 1

⇐⇒ n + 1n + 2

<

(n(n + 2)(n + 1)2

)n

⇐⇒ 1− 1n + 2

<

(1− 1

(n + 1)2

)n

.

Apì thn anisìthta Bernoulli èqoume(

1− 1(n + 1)2

)n

> 1− n

(n + 1)2.

ArkeÐ loipìn na elègxoume ìti

n

(n + 1)2<

1n + 2

,

to opoÐo isqÔei gia k�je n ∈ N.(b) Gia na deÐxoume ìti h (an) eÐnai �nw fragmènh, jewroÔme thn akoloujÐabn =

(1 + 1

n

)n+1. Parathr ste ìti an < bn gia k�je n ∈ N.H (bn) eÐnai gnhsÐwc fjÐnousa: gia na deÐxoume ìti bn > bn+1 gia k�je n ∈ N

parathroÔme ìti(

1 +1n

)n+1

>

(1 +

1n + 1

)n+2

⇐⇒(

n + 1n

)n+1

>

(n + 2n + 1

)n+1n + 2n + 1

⇐⇒ n + 2n + 1

<

((n + 1)2

n(n + 2)

)n+1

⇐⇒ 1 +1

n + 1<

(1 +

1n(n + 2)

)n+1

.

Apì thn anisìthta Bernoulli èqoume(

1 +1

n(n + 2)

)n+1

> 1 +n + 1

n(n + 2).

ArkeÐ loipìn na elègxoume ìti

n + 1n(n + 2)

>1

n + 1,

to opoÐo isqÔei gia k�je n ∈ N.'Epetai ìti an < bn < b1 gia k�je n ∈ N. Dhlad , an < (1 + 1)2 = 4 gia

k�je n ∈ N. EpÐshc, h fjÐnousa akoloujÐa (bn) eÐnai k�tw fragmènh: bn > an >

a1 = 2 gia k�je n ∈ N.Apì to je¸rhma sÔgklishc monìtonwn akolouji¸n, oi (an) kai (bn) sugklinoun.'Eqoun m�lista to Ðdio ìrio: afoÔ bn = an ·

(1 + 1

n

), sumperaÐnoume ìti

limn→∞

bn = limn→∞

an · limn→∞

(1 +

1n

)= lim

n→∞an.

Onom�zoume e to koinì ìrio twn (an) kai (bn). 'Eqoume dh deÐ ìti 2 < e < 4. Giana proseggÐsoume thn tim tou orÐou kalÔtera, parathroÔme ìti, gia par�deigma,an n ≥ 5 tìte a5 < an < e < bn < b5, kai sunep¸c,

2.48832 =(

65

)5

< e <

(65

)6

= 2.985984.

Dhlad , 2 < e < 3. 2

2.5gþ Arq twn kibwtismènwn diasthm�twn

Mia shmantik efarmog tou Jewr matoc 2.5.3 eÐnai h {arq twn kibwtismènwndiasthm�twn}:

Je¸rhma 2.5.5. 'Estw [a1, b1] ⊇ · · · ⊇ [an, bn] ⊇ [an+1, bn+1] ⊇ · · · miafjÐnousa akoloujÐa kleist¸n diasthm�twn. Tìte,

∞⋂n=1

[an, bn] 6= ∅.

An epiplèon bn − an → 0, tìte to sÔnolo⋂∞

n=1[an, bn] perièqei akrib¸c ènanpragmatikì arijmì (eÐnai monosÔnolo).

Apìdeixh. Apì thn [an, bn] ⊇ [an+1, bn+1] èpetai ìti

an ≤ an+1 ≤ bn+1 ≤ bn

gia k�je n ∈ N. Sunep¸c, h (an) eÐnai aÔxousa kai h (bn) eÐnai fjÐnousa.Apì thn [an, bn] ⊆ [a1, b1] blèpoume ìti

a1 ≤ an ≤ bn ≤ b1

gia k�je n ∈ N. Sunep¸c, h (an) eÐnai �nw fragmènh apì ton b1 kai h (bn) eÐnaik�tw fragmènh apì ton a1.

Apì to je¸rhma sÔgklishc monìtonwn akolouji¸n, up�rqoun a, b ∈ R ¸ste

an → a kai bn → b.

AfoÔ an ≤ bn gia k�je n ∈ N, h Prìtash 2.3.11 deÐqnei ìti a ≤ b. EpÐshc, hmonotonÐa twn (an), (bn) dÐnei

an ≤ a kai b ≤ bn

gia k�je n ∈ N, dhlad [a, b] ⊆

∞⋂n=1

[an, bn],

ìpou sumfwnoÔme ìti [a, b] = {a} = {b} an a = b. Eidikìtera,

∞⋂n=1

[an, bn] 6= ∅.

IsqÔei m�lista ìti

[a, b] =∞⋂

n=1

[an, bn].

Pr�gmati, an x ∈ ⋂∞n=1[an, bn] tìte an ≤ x ≤ bn gia k�je n ∈ N, �ra a =

limn an ≤ x ≤ limn bn = b. Dhlad , x ∈ [a, b].Tèloc, an upojèsoume ìti bn − an → 0, èqoume

b− a = limn

bn − limn

an = limn

(bn − an) = 0.

Dhlad , a = b. 'Ara to sÔnolo⋂∞

n=1[an, bn] perièqei akrib¸c ènan pragmatikìarijmì: ton a(= b). 2

Parat rhsh 2.5.6. H upìjesh ìti ta kibwtismèna diast mata tou Jewr -matoc 2.5.5 eÐnai kleist� den mporeÐ na paraleifjeÐ. Gia par�deigma, jewr steta anoikt� diast mata (an, bn) =

(0, 1

n

). 'Eqoume

(0, 1) ⊇(

0,12

)⊇ · · · ⊇

(0,

1n

)⊇

(0,

1n + 1

)⊇ · · · ,

ìmwc∞⋂

n=1

(0,

1n

)= ∅.

Ali¸c, ja up rqe x > 0 pou ja ikanopoioÔse thn x < 1n gia k�je n ∈ N. Autì

eÐnai adÔnato, lìgw thc Arqim deiac idiìthtac.

2.5dþ Anadromikèc akoloujÐec

KleÐnoume aut n thn Par�grafo me èna par�deigma anadromik c akoloujÐac. Hteqnik pou qrhsimopoioÔme gia th melèth thc sÔgklishc anadromik¸n akolou-ji¸n basÐzetai suqn� sto je¸rhma sÔgklishc monìtonwn akolouji¸n.

Par�deigma 2.5.7. JewroÔme thn akoloujÐa (an) pou èqei pr¸to ìro tona1 = 1 kai ikanopoieÐ thn anadromik sqèsh an+1 =

√1 + an gia n ≥ 1. Ja

deÐxoume ìti h (an) sugklÐnei ston arijmì 1+√

52 .

Apìdeixh. Apì ton trìpo orismoÔ thc (an) eÐnai fanerì ìti ìloi oi ìroi thc eÐnaijetikoÐ (deÐxte to austhr� me epagwg ).

Ac upojèsoume ìti èqoume katafèrei na deÐxoume ìti an → a gia k�poiona ∈ R. Tìte,

an+1 → a kai√

1 + an →√

1 + a.

AfoÔ an+1 =√

1 + an, apì th monadikìthta tou orÐou o a prèpei na ikanopoieÐthn exÐswsh a =

√1 + a, dhlad a2 − a− 1 = 0. Sunep¸c,

a =1 +

√5

2 a =

1−√52

.

'Omwc, to ìrio thc (an), an up�rqei, eÐnai mh arnhtikì. 'Ara, a = 1+√

52 .

Mènei na deÐxoume thn Ôparxh tou orÐou. ParathroÔme ìti a2 =√

2 > 1 = a1.Mia idèa eÐnai loipìn na deÐxoume ìti eÐnai aÔxousa kai �nw fragmènh. Tìte, apìto je¸rhma sÔgklishc monìtonwn akolouji¸n, h (an) sugklinei (kai to ìrio thceÐnai o 1+

√5

2 ).(a) DeÐqnoume me epagwg ìti an+1 ≥ an gia k�je n ∈ N. 'Eqoume dh elègxeiìti a2 > a1. Upojètontac ìti am+1 ≥ am, paÐrnoume

am+2 =√

1 + am+1 ≥√

1 + am = am+1,

dhlad èqoume deÐxei to epagwgikì b ma.(b) Tèloc, deÐqnoume me epagwg ìti h (an) eÐnai �nw fragmènh. Apì th stigm pou èqoume deÐxei ìti h (an) eÐnai aÔxousa, ja èprepe na mporoÔme na deÐxoumeìti k�je pragmatikìc arijmìc megalÔteroc Ðsoc apì to {upoy fio ìrio} 1+

√5

2

eÐnai �nw fr�gma thc (an). Gia par�deigma, mporoÔme eÔkola na doÔme ìti an < 2gia k�je n ∈ N. 'Eqoume a1 = 1 < 2 kai an am < 2 tìte am+1 =

√1 + am <√

1 + 2 =√

3 < 2. 2

2.6 Je¸rhma Bolzano-Weierstrass

2.6aþ UpakoloujÐec

Orismìc 2.6.1. 'Estw (an) mia akoloujÐa pragmatik¸n arijm¸n. H akolou-jÐa (bn) lègetai upakoloujÐa thc (an) an up�rqei gnhsÐwc aÔxousa akoloujÐafusik¸n arijm¸n k1 < k2 < · · · < kn < kn+1 < · · · ¸ste

bn = akn gia k�je n ∈ N.

Me �lla lìgia, oi ìroi thc (bn) eÐnai oi ak1 , ak2 , . . . , akn , . . ., ìpou k1 < k2 <

· · · < kn < kn+1 < · · · . Genik�, mia akoloujÐa èqei pollèc (sun jwc �peirec topl joc) diaforetikèc upakoloujÐec.

ParadeÐgmata 2.6.2. 'Estw (an) mia akoloujÐa pragmatik¸n arijm¸n.(a) H upakoloujÐa (a2n) twn {�rtiwn ìrwn} thc (an) èqei ìrouc touc

a2, a4, a6, . . . .

Ed¸, kn = 2n.(b) H upakoloujÐa (a2n−1) twn {peritt¸n ìrwn} thc (an) èqei ìrouc touc

a1, a3, a5, . . . .

Ed¸, kn = 2n− 1.(g) H upakoloujÐa (an2) thc (an) èqei ìrouc touc

a1, a4, a9, . . . .

Ed¸, kn = n2.(d) K�je telikì tm ma (am, am+1, am+2, . . .) thc (an) eÐnai upakoloujÐa thc (an).Ed¸, kn = m + n− 1.

2.6 Jewrhma Bolzano-Weierstrass · 51

Parat rhsh 2.6.3. 'Estw (kn) mia gnhsÐwc aÔxousa akoloujÐa fusik¸narijm¸n. Tìte, kn ≥ n gia k�je n ∈ N.Apìdeixh. Me epagwg : afoÔ o k1 eÐnai fusikìc arijmìc, eÐnai fanerì ìti k1 ≥ 1.Gia to epagwgikì b ma upojètoume ìti km ≥ m. AfoÔ h (kn) eÐnai gnhsÐwcaÔxousa, èqoume km+1 > km, �ra km+1 > m. AfoÔ oi km+1 kai m eÐnai fusikoÐarijmoÐ, èpetai ìti km+1 ≥ m + 1. 2

H epìmenh Prìtash deÐqnei ìti an mia akoloujÐa sugklÐnei se pragmatikìarijmì tìte ìlec oi upakoloujÐec thc eÐnai sugklÐnousec kai sugklÐnoun stonÐdio pragmatikì arijmì.

Prìtash 2.6.4. An an → a tìte gia k�je upakoloujÐa (akn) thc (an) isqÔei

akn→ a.

Apìdeixh. 'Estw ε > 0. AfoÔ an → a, up�rqei n0 = n0(ε) ∈ N me thn ex cidiìthta:

Gia k�je m ≥ n0 isqÔei |am − a| < ε.

Apì thn Parat rhsh 2.6.3 gia k�je n ≥ n0 èqoume kn ≥ n ≥ n0. Jètontacloipìn m = kn sthn prohgoÔmenh sqèsh, paÐrnoume:

Gia k�je n ≥ n0 isqÔei |akn − a| < ε.

Autì apodeiknÔei ìti akn → a: gia to tuqìn ε > 0 br kame n0 ∈ N ¸ste ìloi oiìroi akn0

, akn0+1 , . . . thc (akn) na an koun sto (a− ε, a + ε). 2

Parat rhsh 2.6.5. H prohgoÔmenh Prìtash eÐnai polÔ qr simh an jèloumena deÐxoume ìti mia akoloujÐa (an) den sugklÐnei se kanènan pragmatikì arijmì.ArkeÐ na broÔme dÔo upakoloujÐec thc (an) oi opoÐec na èqoun diaforetik� ìria.

Gia par�deigma, ac jewr soume thn (an) = (−1)n. Tìte, a2n = (−1)2n =1 → 1 kai a2n−1 = (−1)2n−1 = −1 → −1.

Ac upojèsoume ìti an → a. Oi (a2n) kai (a2n−1) eÐnai upakoloujÐec thc(an), prèpei loipìn na isqÔei a2n → a kai a2n−1 → a. Apì th monadikìthta touorÐou thc (a2n) paÐrnoume a = 1 kai apì th monadikìthta tou orÐou thc (a2n−1)paÐrnoume a = −1. Dhlad , 1 = −1. Katal xame se �topo, �ra h (an) densugklÐnei.

2.6bþ Je¸rhma Bolzano-Weierstrass

Je¸rhma 2.6.6 (Bolzano-Weierstrass). K�je fragmènh akoloujÐa èqeiupakoloujÐa pou sugklÐnei se pragmatikì arijmì.

Ja d¸soume dÔo apodeÐxeic autoÔ tou Jewr matoc. H pr¸th basÐzetai stogegonìc ìti k�je monìtonh kai fragmènh akoloujÐa sugklÐnei. Gia na broÔme su-gklinousa upakoloujÐa miac fragmènhc akoloujÐac arkeÐ na broÔme mia monìtonhupakoloujÐa thc. To teleutaÐo isqÔei entel¸c genik�, ìpwc deÐqnei to epìmenoJe¸rhma:

Je¸rhma 2.6.7. K�je akoloujÐa èqei toul�qiston mÐa monìtonh upakoloujÐa.

Apìdeixh. Ja qreiastoÔme thn ènnoia tou shmeÐou koruf c miac akoloujÐac.

Orismìc 2.6.8. 'Estw (an) mia akoloujÐa pragmatik¸n arijm¸n. Lème ìti oam eÐnai shmeÐo koruf c thc (an) an am ≥ an gia k�je n ≥ m.[Gia na exoikeiwjeÐte me ton orismì elègxte ta ex c. An h (an) eÐnai fjÐnousatìte k�je ìroc thc eÐnai shmeÐo koruf c thc. An h (an) eÐnai gnhsÐwc aÔxousatìte den èqei kanèna shmeÐo koruf c.]

'Estw (an) mia akoloujÐa pragmatik¸n arijm¸n. DiakrÐnoume dÔo peript¸-seic:(a) H (an) èqei �peira to pl joc shmeÐa koruf c. Tìte, up�rqoun fusikoÐ arijmoÐk1 < k2 < · · · < kn < kn+1 < · · · ¸ste ìloi oi ìroi ak1 , . . . , akn , . . . na eÐnaishmeÐa koruf c thc (an) (exhg ste giatÐ). AfoÔ kn < kn+1 gia k�je n ∈ N, h(akn) eÐnai upakoloujÐa thc (an). Apì ton orismì tou shmeÐou koruf c blèpoumeìti gia k�je n ∈ N isqÔei akn

≥ akn+1 (èqoume kn+1 > kn kai o akneÐnai shmeÐo

koruf c thc (an)). Dhlad ,

ak1 ≥ ak2 ≥ · · · ≥ akn ≥ akn+1 ≥ · · · .

'Ara, h upakoloujÐa (akn) eÐnai fjÐnousa.(b) H (an) èqei peperasmèna to pl joc shmeÐa koruf c. Tìte, up�rqei N ∈ Nme thn ex c idiìthta: an m ≥ N tìte o am den eÐnai shmeÐo koruf c thc (an)(p�rte N = k + 1 ìpou ak to teleutaÐo shmeÐo koruf c thc (an) N = 1 an denup�rqoun shmeÐa koruf c).

Me b�sh ton orismì tou shmeÐou koruf c autì shmaÐnei ìti: an m ≥ N tìteup�rqei n > m ¸ste an > am.

Efarmìzoume diadoqik� to parap�nw. Jètoume k1 = N kai brÐskoume k2 > k1

¸ste ak2 > ak1 . Katìpin brÐskoume k3 > k2 ¸ste ak3 > ak2 kai oÔtw kajex c.Up�rqoun dhlad k1 < k2 < · · · < kn < kn+1 < · · · ¸ste

ak1 < ak2 < · · · < akn < akn+1 < · · · .

Tìte, h (akn) eÐnai gnhsÐwc aÔxousa upakoloujÐa thc (an). 2

MporoÔme t¸ra na apodeÐxoume to Je¸rhma Bolzano-Weierstrass.

Apìdeixh tou Jewr matoc 2.6.6. 'Estw (an) fragmènh akoloujÐa. Apì to Je¸-rhma 2.6.7 h (an) èqei monìtonh upakoloujÐa (akn). H (akn) eÐnai monìtonh kaifragmènh, sunep¸c sugklÐnei se pragmatikì arijmì. 2

2.6gþ DeÔterh apìdeixh me qr sh thc arq c tou kibwtismoÔ

H deÔterh apìdeixh tou Jewr matoc Bolzano-Weierstrass qrhsimopoieÐ thn arq twn kibwtismènwn diasthm�twn (Je¸rhma 2.5.5). 'Estw (an) mia fragmènh ako-loujÐa pragmatik¸n arijm¸n. Tìte, up�rqei kleistì di�sthma [b1, c1] sto opoÐoan koun ìloi oi ìroi an.

QwrÐzoume to [b1, c1] se dÔo diadoqik� diast mata m kouc c1−b12 : ta

[b1,

b1+c12

]

kai[

b1+c12 , c1

]. K�poio apì aut� ta dÔo diast mata perièqei �peirouc to pl joc

ìrouc thc (an). PaÐrnontac san [b2, c2] autì to upodi�sthma tou [b1, c1] èqoumedeÐxei to ex c.

2.7 Akoloujiec Cauchy · 53

Up�rqei kleistì di�sthma [b2, c2] ⊂ [b1, c1] to opoÐo perièqei �peiroucìrouc thc (an) kai èqei m koc

c2 − b2 =c1 − b1

2.

SuneqÐzoume me ton Ðdio trìpo: qwrÐzoume to [b2, c2] se dÔo diadoqik� dia-st mata m kouc c2−b2

2 : ta[b2,

b2+c22

]kai

[b2+c2

2 , c2

]. AfoÔ to [b2, c2] perièqei

�peirouc ìrouc thc (an), k�poio apì aut� ta dÔo diast mata perièqei �peirouc topl joc ìrouc thc (an). PaÐrnontac san [b3, c3] autì to upodi�sthma tou [b2, c2]èqoume deÐxei to ex c.

Up�rqei kleistì di�sthma [b3, c3] ⊂ [b2, c2] to opoÐo perièqei �peiroucìrouc thc (an) kai èqei m koc

c3 − b3 =c2 − b2

2=

c1 − b1

22.

SuneqÐzontac me ton Ðdio trìpo orÐzoume akoloujÐa ([bm, cm]) kleist¸n dia-sthm�twn pou ikanopoieÐ ta ex c:

(i) Gia k�je m ∈ N isqÔei [bm+1, cm+1] ⊂ [bm, cm].

(ii) Gia k�je m ∈ N isqÔei cm − bm = (c1 − b1)/2m−1.

(iii) Gia k�je m ∈ N up�rqoun �peiroi ìroi thc (an) sto [bm, cm].

Qrhsimopoi¸ntac thn trÐth sunj kh, mporoÔme na broÔme upakoloujÐa (akm)thc (an) me thn idiìthta: gia k�je m ∈ N isqÔei akm ∈ [bm, cm]. Pr�gmati,up�rqei k1 ∈ N ¸ste ak1 ∈ [b1, c1] � gia thn akrÐbeia, ìloi oi ìroi thc (an)brÐskontai sto [b1, c1]. T¸ra, afoÔ to [b2, c2] perièqei �peirouc ìrouc thc (an),k�poioc apì autoÔc èqei deÐkth megalÔtero apì k1. Dhlad , up�rqei k2 > k1

¸ste ak2 ∈ [b2, c2]. Me ton Ðdio trìpo, an èqoun oristeÐ k1 < · · · < km ¸steaks ∈ [bs, cs] gia k�je s = 1, . . . , m, mporoÔme na broÔme km+1 > km ¸steakm+1 ∈ [bm+1, cm+1] (diìti, to [bm+1, cm+1] perièqei �peirouc ìrouc thc (an)).'Etsi, orÐzetai mia upakoloujÐa (akm) thc (an) pou ikanopoieÐ to zhtoÔmeno.

Ja deÐxoume ìti h (akm) sugklÐnei. Apì thn arq twn kibwtismènwn diasth-m�twn up�rqei monadikìc a ∈ R o opoÐoc an kei se ìla ta kleist� diast mata[bm, cm]. JumhjeÐte ìti

limm→∞

bm = a = limm→∞

cm.

AfoÔ bm ≤ akm ≤ cm gia k�je m, to krit rio twn isosugklinous¸n akolouji¸ndeÐqnei ìti akm → a. 2

2.7 AkoloujÐec Cauchy

O orismìc thc akoloujÐac Cauchy èqei san afethrÐa thn ex c parat rhsh: acupojèsoume ìti an → a. Tìte, oi ìroi thc (an) eÐnai telik� {kont�} sto a,

�ra eÐnai telik� kai {metaxÔ touc kont�}. Gia na ekfr�soume austhr� aut thnparat rhsh, ac jewr soume tuqìn ε > 0. Up�rqei n0 = n0(ε) ∈ N ¸ste giak�je n ≥ n0 na isqÔei |an − a| < ε

2 . Tìte, gia k�je n,m ≥ n0 èqoume

|an − am| ≤ |an − a|+ |a− am| < ε

2+

ε

2= ε.

Orismìc 2.7.1. Mia akoloujÐa (an) lègetai akoloujÐa Cauchy ( basik akoloujÐa) an gia k�je ε > 0 up�rqei n0 = n0(ε) ∈ N ¸ste:

an m,n ≥ n0(ε), tìte |an − am| < ε.

Parat rhsh 2.7.2. An h (an) eÐnai akoloujÐa Cauchy, tìte gia k�je ε > 0up�rqei n0 = n0(ε) ∈ N ¸ste

an n ≥ n0(ε), tìte |an − an+1| < ε.

To antÐstrofo den isqÔei: an, apì k�poion deÐkth kai pèra, diadoqikoÐ ìroi eÐnaikont�, den èpetai ìti h akoloujÐa eÐnai Cauchy. Gia par�deigma, jewr ste thn

an = 1 +1√2

+ · · ·+ 1√n

.

Tìte,

|an+1 − an| = 1√n + 1

→ 0

ìtan n →∞, ìmwc

|a2n − an| = 1√n + 1

+ · · ·+ 1√2n

>

√n√2→ +∞

ìtan n →∞, ap� ìpou blèpoume ìti h (an) den eÐnai akoloujÐa Cauchy (exhg stegiatÐ).

Skopìc mac eÐnai na deÐxoume ìti mia akoloujÐa pragmatik¸n arijm¸n eÐnaisugklÐnousa an kai mìno an eÐnai akoloujÐa Cauchy. H apìdeixh gÐnetai se trÐab mata.

Prìtash 2.7.3. K�je akoloujÐa Cauchy eÐnai fragmènh.

Apìdeixh. 'Estw (an) akoloujÐa Cauchy. P�rte ε = 1 > 0 ston orismì: up�rqein0 ∈ N ¸ste |an − am| < 1 gia k�je n,m ≥ n0. Eidikìtera, |an − an0 | < 1 giak�je n > n0. Dhlad ,

|an| < 1 + |an0 | gia k�je n > n0.

Jètoume M = max{|a1|, . . . , |an0 |, 1 + |an0 |} kai eÔkola epalhjeÔoume ìti

|an| ≤ M

gia k�je n ∈ N. 2

2.7 Akoloujiec Cauchy · 55

Prìtash 2.7.4. An mia akoloujÐa Cauchy (an) èqei sugklÐnousa upakoloujÐa,tìte h (an) sugklÐnei.

Apìdeixh. Upojètoume ìti h (an) eÐnai akoloujÐa Cauchy kai ìti h upakoloujÐa(akn

) sugklÐnei sto a ∈ R. Ja deÐxoume ìti an → a.

'Estw ε > 0. AfoÔ akn → a, up�rqei n1 ∈ N ¸ste: gia k�je n ≥ n1,

|akn− a| < ε

2.

AfoÔ h (an) eÐnai akoloujÐa Cauchy, up�rqei n2 ∈ N ¸ste: gia k�je n,m ≥ n2

|an − am| < ε

2.

Jètoume n0 = max{n1, n2}. 'Estw n ≥ n0. Tìte kn ≥ n ≥ n0 ≥ n1, �ra

|akn− a| < ε

2.

EpÐshc kn, n ≥ n0 ≥ n2, �ra

|akn − an| < ε

2.

'Epetai ìti|an − a| ≤ |an − akn |+ |akn − a| < ε

2+

ε

2= ε.

Dhlad , |an − a| < ε gia k�je n ≥ n0. Autì shmaÐnei ìti an → a. 2

Je¸rhma 2.7.5. Mia akoloujÐa (an) sugklÐnei an kai mìno an eÐnai akoloujÐaCauchy.

Apìdeixh. H mÐa kateÔjunsh apodeÐqthke sthn eisagwg aut c thc paragr�fou:an upojèsoume ìti an → a kai an jewr soume tuqìn ε > 0, up�rqei n0 = n0(ε) ∈N ¸ste gia k�je n ≥ n0 na isqÔei |an−a| < ε

2 . Tìte, gia k�je n, m ≥ n0 èqoume

|an − am| ≤ |an − a|+ |a− am| < ε

2+

ε

2= ε.

'Ara, h (an) eÐnai akoloujÐa Cauchy.

Gia thn antÐstrofh kateÔjunsh: èstw (an) akoloujÐa Cauchy. Apì thn Prìta-sh 2.7.3, h (an) eÐnai fragmènh. Apì to Je¸rhma Bolzano-Weierstrass, h (an)èqei sugklÐnousa upakoloujÐa. Tèloc, apì thn Prìtash 2.7.4 èpetai ìti h (an)sugklÐnei. 2

Autì to krit rio sÔgklishc eÐnai polÔ qr simo. Pollèc forèc jèloume naexasfalÐsoume thn Ôparxh orÐou gia mia akoloujÐa qwrÐc na mac endiafèrei htim tou orÐou. ArkeÐ na deÐxoume ìti h akoloujÐa eÐnai Cauchy, dhlad ìti oiìroi thc eÐnai {kont�} gia meg�louc deÐktec, k�ti pou den apaiteÐ na mantèyoumeek twn protèrwn poiì eÐnai to ìrio. AntÐjeta, gia na doulèyoume me ton orismìtou orÐou, prèpei dh na xèroume poiì eÐnai to upoy fio ìrio (sugkrÐnete toucdÔo orismoÔc: {an → a} kai {(an) akoloujÐa Cauchy}.)

2.8 *An¸tero kai kat¸tero ìrio akoloujÐac

Skopìc mac se aut n thn Par�grafo eÐnai na melet soume pio prosektik� ticupakoloujÐec miac fragmènhc akoloujÐac. JumhjeÐte ìti an h akoloujÐa (an)sugklÐnei se k�poion pragmatikì arijmì a tìte h kat�stash eÐnai polÔ apl .An (akn) eÐnai tuqoÔsa upakoloujÐa thc (an), tìte akn → a. Dhlad , ìlec oiupakoloujÐec miac sugklÐnousac akoloujÐac sugklÐnoun kai m�lista sto ìriothc akoloujÐac.

Orismìc 2.8.1. 'Estw (an) mia akoloujÐa. Lème ìti o x ∈ R eÐnai shmeÐosuss¸reushc ( upakoloujiakì ìrio) thc (an) an up�rqei upakoloujÐa (akn

)thc (an) ¸ste akn

→ x.

Ta shmeÐa suss¸reushc miac akoloujÐac qarakthrÐzontai apì to epìmeno L mma.

L mma 2.8.2. O x eÐnai shmeÐo suss¸reushc thc (an) an kai mìno an gia k�jeε > 0 kai gia k�je m ∈ N up�rqei n ≥ m ¸ste |an − x| < ε.

Apìdeixh. Upojètoume pr¸ta ìti o x eÐnai shmeÐo suss¸reushc thc (an). Up�r-qei loipìn upakoloujÐa (akn) thc (an) ¸ste akn → x.

'Estw ε > 0 kai m ∈ N. Up�rqei n0 ∈ N ¸ste |akn − x| < ε gia k�jen ≥ n0. JewroÔme ton n1 = max{m,n0}. Tìte kn1 ≥ n1 ≥ m kai n1 ≥ n0, �ra|akn1

− x| < ε.AntÐstrofa: PaÐrnoume ε = 1 kai m = 1. Apì thn upìjesh up�rqei k1 ≥ 1 ¸ste|ak1 − x| < 1. Sth sunèqeia paÐrnoume ε = 1

2 kai m = k1 + 1. Efarmìzontac thnupìjesh brÐskoume k2 ≥ k1 + 1 > k1 ¸ste |ak2 − x| < 1

2 .Epagwgik� brÐskoume k1 < k2 < · · · < kn < · · · ¸ste

|akn − x| < 1n

(k�nete mìnoi sac to epagwgikì b ma). EÐnai fanerì ìti akn → x. 2

'Estw (an) mia fragmènh akoloujÐa. Dhlad , up�rqei M > 0 ¸ste |an| ≤ M

gia k�je n ∈ N. JewroÔme to sÔnolo

K = {x ∈ R : x eÐnai shmeÐo suss¸reushc thc (an)}.

1. To K eÐnai mh kenì. Apì to Je¸rhma Bolzano-Weierstrass up�rqei toul�qi-ston mÐa upakoloujÐa (akn) thc (an) pou sugklÐnei se pragmatikì arijmì. Toìrio thc (akn) eÐnai ex orismoÔ stoiqeÐo tou K.2. To K eÐnai fragmèno. An x ∈ K, up�rqei akn → x kai afoÔ −M ≤ akn ≤ M

gia k�je n, èpetai ìti −M ≤ x ≤ M .

Apì to axÐwma thc plhrìthtac prokÔptei ìti up�rqoun ta supK kai inf K. Toepìmeno L mma deÐqnei ìti to K èqei mègisto kai el�qisto stoiqeÐo.

L mma 2.8.3. 'Estw (an) fragmènh akoloujÐa kai

K = {x ∈ R : x eÐnai shmeÐo suss¸reushc thc (an)}.

Tìte, sup K ∈ K kai inf K ∈ K.

2.8 *Anwtero kai katwtero orio akoloujiac · 57

Apìdeixh. 'Estw a = sup K. Jèloume na deÐxoume ìti o a eÐnai shmeÐo suss¸reu-shc thc (an), kai sÔmfwna me to L mma 2.8.2 arkeÐ na doÔme ìti gia k�je ε > 0kai gia k�je m ∈ N up�rqei n ≥ m ¸ste |an − a| < ε.

'Estw ε > 0 kai m ∈ N. AfoÔ a = sup K, up�rqei x ∈ K ¸ste a− ε2 < x ≤ a.

O x eÐnai shmeÐo suss¸reushc thc (an), �ra up�rqei n ≥ m ¸ste |an − x| < ε2 .

Tìte,|an − a| ≤ |an − x|+ |x− a| < ε

2+

ε

2= ε.

Me an�logo trìpo deÐqnoume ìti inf K ∈ K. 2

Orismìc 2.8.4. 'Estw (an) mia fragmènh akoloujÐa. An

K = {x ∈ R : x eÐnai shmeÐo suss¸reushc thc (an)},

orÐzoume

(i) lim sup an = sup K, to an¸tero ìrio thc (an),

(ii) lim inf an = inf K to kat¸tero ìrio thc (an).

Me b�sh to L mma 2.8.3, to lim sup an eÐnai to mègisto stoiqeÐo kai tolim inf an eÐnai to el�qisto stoiqeÐo tou K antÐstoiqa:

Je¸rhma 2.8.5. 'Estw (an) fragmènh akoloujÐa. To lim sup an eÐnai o me-galÔteroc pragmatikìc arijmìc x gia ton opoÐo up�rqei upakoloujÐa (akn) thc(an) me akn → x. To lim inf an eÐnai o mikrìteroc pragmatikìc arijmìc y gia tonopoÐo up�rqei upakoloujÐa (aln) thc (an) me aln → y. 2

To an¸tero kai to kat¸tero ìrio miac fragmènhc akoloujÐac perigr�fontaimèsw twn perioq¸n touc wc ex c:

Je¸rhma 2.8.6. 'Estw (an) fragmènh akoloujÐa pragmatik¸n arijm¸n kaièstw x ∈ R. Tìte,(1) x ≤ lim sup an an kai mìno an: gia k�je ε > 0 to sÔnolo {n ∈ N : x−ε < an}eÐnai �peiro.(2) x ≥ lim sup an an kai mìno an: gia k�je ε > 0 to sÔnolo {n ∈ N : x+ε < an}eÐnai peperasmèno.(3) x ≥ lim inf an an kai mìno an: gia k�je ε > 0 to sÔnolo {n ∈ N : an < x+ ε}eÐnai �peiro.(4) x ≤ lim inf an an kai mìno an: gia k�je ε > 0 to sÔnolo {n ∈ N : an < x− ε}eÐnai peperasmèno.(5) x = lim sup an an kai mìno an: gia k�je ε > 0 to {n ∈ N : x− ε < an} eÐnai�peiro kai to {n ∈ N : x + ε < an} eÐnai peperasmèno.(6) x = lim inf an an kai mìno an: gia k�je ε > 0 to {n ∈ N : an < x + ε} eÐnai�peiro kai to {n ∈ N : an < x− ε} eÐnai peperasmèno.

Apìdeixh. (1:⇒) 'Estw ε > 0. Up�rqei upakoloujÐa (akn) thc (an) me akn →lim sup an, �ra up�rqei n0 ¸ste gia k�je n ≥ n0

akn > lim sup an − ε ≥ x− ε.

'Epetai ìti to {n : an > x− ε} eÐnai �peiro.

(2:⇒) 'Estw ε > 0. An up rqan k1 < k2 < · · · < kn < · · · me akn> x + ε,

tìte h (akn) ja eÐqe sugklÐnousa upakoloujÐa (alkn

) (apì to Je¸rhma Bolzano-Weierstrass) me alkn

→ y ≥ x + ε. 'Omwc tìte ja eÐqame

lim sup an ≥ y ≥ x + ε ≥ lim sup an + ε

to opoÐo eÐnai �topo. 'Ara, to {n : an > x + ε} eÐnai peperasmèno.

(1: ⇐) 'Estw ìti x > lim sup an. Tìte up�rqei ε > 0 ¸ste an y = x−ε na èqoumex > y > lim sup an. Apì thn upìjes mac, to {n ∈ N : y < an} eÐnai �peiro.'Omwc y > lim sup an opìte apì thn (2: ⇒) to sÔnolo {n ∈ N : y < an} eÐnaipeperasmèno (gr�yte y = lim sup an + ε1 gia k�poio ε1 > 0). Oi dÔo isqurismoÐèrqontai se antÐfash.

(2:⇐) 'Omoia, upojètoume ìti x < lim sup an kai brÐskoume y ¸ste x < y <

lim sup an. AfoÔ y > x, sumperaÐnoume ìti to {n ∈ N : y < an} eÐnai peperasmèno(aut eÐnai h upìjes mac) kai afoÔ y < lim sup an sumperaÐnoume ìti to {n ∈N : y < an} eÐnai �peiro (apì thn (1:⇒).) 'Etsi katal goume se �topo.

H (5) eÐnai �mesh sunèpeia twn (1) kai (2).

Gia tic (3), (4) kai (6) ergazìmaste ìmoia. 2

Mia enallaktik perigraf twn lim sup an kai lim inf an dÐnetai apì to epìmenoje¸rhma:

Je¸rhma 2.8.7. 'Estw (an) fragmènh akoloujÐa.

(a) Jètoume bn = sup{ak : k ≥ n}. Tìte, lim sup an = inf{bn : n ∈ N}.(b) Jètoume γn = inf{ak : k ≥ n}. Tìte, lim inf an = sup{γn : n ∈ N}.

Apìdeixh. DeÐqnoume pr¸ta ìti oi arijmoÐ inf{bn : n ∈ N} kai sup{γn : n ∈ N}orÐzontai kal�:

Gia k�je n ∈ N, isqÔei γn ≤ an ≤ bn (exhg ste giatÐ). EpÐshc, h (bn)eÐnai fjÐnousa, en¸ h (an) eÐnai aÔxousa (exhg ste giatÐ). AfoÔ h (an) eÐnaifragmènh, èpetai ìti h (bn) eÐnai fjÐnousa kai k�tw fragmènh, en¸ h (γn) eÐnaiaÔxousa kai �nw fragmènh. Apì to je¸rhma sÔgklishc monìtonwn akolouji¸nsumperaÐnoume ìti bn → inf{bn : n ∈ N} := b kai γn → sup{γn : n ∈ N} := γ.

Ja deÐxoume ìti lim sup an = b. Apì to L mma 2.8.3 up�rqei upakoloujÐa(akn) thc (an) me akn → lim sup an. 'Omwc, akn ≤ bkn kai bkn → b (exhg stegiatÐ). 'Ara,

lim sup an = lim akn ≤ lim bkn = b.

Gia thn antÐstrofh anisìthta deÐqnoume ìti o b eÐnai shmeÐo suss¸reushc thc(an). 'Estw ε > 0 kai èstw m ∈ N. Up�rqei n ≥ m ¸ste |b − bn| < ε

2 . All�,bn = sup{ak : k ≥ n}, �ra up�rqei k ≥ n ≥ m ¸ste bn ≥ ak > bn − ε

2 dhlad |bn − ak| < ε

2 . 'Epetai ìti

|b− ak| ≤ |b− bn|+ |bn − ak| < ε

2+

ε

2= ε.

2.9 *Pararthma: suzhthsh gia to axiwma thc plhrothtac · 59

Apì to L mma 2.8.2 o b eÐnai shmeÐo suss¸reushc thc (an), kai sunep¸c, b ≤lim sup an.

Me an�logo trìpo deÐqnoume ìti lim inf an = γ. 2

KleÐnoume me ènan qarakthrismì thc sÔgklishc gia fragmènec akoloujÐec.

Je¸rhma 2.8.8. 'Estw (an) fragmènh akoloujÐa. H (an) sugklÐnei an kaimìno an lim sup an = lim inf an.

Apìdeixh. An an → a tìte gia k�je upakoloujÐa (akn) thc (an) èqoume akn

→ a.Epomènwc, o a eÐnai to monadikì shmeÐo suss¸reushc thc (an). 'Eqoume K = {a},�ra

lim sup an = lim inf an = a.

AntÐstrofa: èstw ε > 0. Apì to Je¸rhma 2.8.6 o arijmìc a = lim sup an =lim inf an èqei thn ex c idiìthta:

Ta sÔnola {n ∈ N : an < a − ε} kai {n ∈ N : an > a + ε} eÐnaipeperasmèna.

Dhlad , to sÔnolo{n ∈ N : |an − a| > ε}

eÐnai peperasmèno. IsodÔnama, up�rqei n0 ∈ N me thn idiìthta: gia k�je n ≥ n0,

|an − a| ≤ ε.

AfoÔ to ε > 0 tan tuqìn, èpetai ìti an → a. 2

Parat rhsh 2.8.9. Ac upojèsoume ìti h akoloujÐa (an) den eÐnai fragmènh.An h (an) den eÐnai �nw fragmènh, tìte up�rqei upakoloujÐa (akn) thc (an) ¸steakn → +∞ (�skhsh). Me �lla lìgia, o +∞ eÐnai {shmeÐo suss¸reushc} thc(an). Se aut n thn perÐptwsh eÐnai logikì na orÐsoume lim sup an = +∞. Ente-l¸c an�loga, an h (an) den eÐnai k�tw fragmènh, tìte up�rqei upakoloujÐa (akn)thc (an) ¸ste akn → −∞ (�skhsh). Dhlad , o −∞ eÐnai {shmeÐo suss¸reushc}thc (an). Tìte, orÐzoume lim inf an = −∞.

2.9 *Par�rthma: suz thsh gia to axÐwma thc plhrì-thtac

Sto Kef�laio 1 deqt kame ìti to R eÐnai èna diatetagmèno s¸ma pou ikanopoieÐ toaxÐwma thc plhrìthtac: k�je mh kenì, �nw fragmèno uposÔnolì tou èqei el�qi-sto �nw fr�gma. Qrhsimopoi¸ntac thn Ôparxh supremum deÐxame thn Arqim deiaidiìthta:

(∗) An a ∈ R kai ε > 0, up�rqei n ∈ N ¸ste nε > a.

Qrhsimopoi¸ntac kai p�li to axÐwma thc plhrìthtac, deÐxame ìti k�je mo-nìtonh kai fragmènh akoloujÐa sugklÐnei. San sunèpeia p rame to Je¸rhmaBolzano-Weierstrass: k�je fragmènh akoloujÐa èqei sugklÐnousa upakoloujÐ-a. Autì me th seir� tou mac epètreye na deÐxoume thn {idiìthta Cauchy} twnpragmatik¸n arijm¸n:

(∗∗) K�je akoloujÐa Cauchy pragmatik¸n arijm¸n sugklÐnei se prag-matikì arijmì.

Se aut n thn par�grafo ja deÐxoume ìti to axÐwma thc plhrìthtac eÐnai logik sunèpeia twn (∗) kai (∗∗). An dhlad deqtoÔme to R san èna diatetagmèno s¸mapou èqei thn Arqim deia idiìthta kai thn idiìthta Cauchy, tìte mporoÔme naapodeÐxoume to {axÐwma thc plhrìthtac} san je¸rhma:

Je¸rhma 2.9.1. 'Estw R∗ èna diatetagmèno s¸ma me tic akìloujec idiìthtec:1. An a ∈ R∗ kai ε ∈ R∗, ε > 0, tìte up�rqei n ∈ N ¸ste nε > a.2. K�je akoloujÐa Cauchy stoiqeÐwn tou R∗ sugklÐnei se stoiqeÐo tou R∗.

Tìte, k�je mh kenì kai �nw fragmèno A ⊂ R∗ èqei el�qisto �nw fr�gma.

ShmeÐwsh: To sÔnolo N, oi akoloujÐec Cauchy, to el�qisto �nw fr�gma enìcsunìlou klp. orÐzontai akrib¸c ìpwc kai sto R: N eÐnai to mikrìtero epagwgikìuposÔnolo tou R∗, h akoloujÐa (an) eÐnai akoloujÐa Cauchy an gia k�je ε > 0up�rqei n0(ε) ∈ N ¸ste |an − am| < ε an n, m ≥ n0(ε), kai oÔtw kajex c.

Apìdeixh: 'Estw A mh kenì kai �nw fragmèno uposÔnolo tou R∗.Xekin�me me tuqìn stoiqeÐo a0 ∈ A (up�rqei afoÔ A 6= εmptyset). 'Estw b �nwfr�gma tou A. Apì thn sunj kh 1, up�rqei k ∈ N gia ton opoÐo a0 + k > b.Dhlad , up�rqei fusikìc k me thn idiìthta

gia k�je a ∈ A, a < a0 + k.

Apì thn arq tou elaqÐstou (pou eÐnai sunèpeia thc 1) èpetai ìti up�rqei el�qi-stoc tètoioc fusikìc. Ac ton poÔme k1. Tìte,

• Gia k�je a ∈ A isqÔei a < a0 + k1.

• Up�rqei a1 ∈ A ¸ste a0 + (k1 − 1) ≤ a1.

Epagwgik� ja broÔme a0 ≤ a1 ≤ . . . ≤ an ≤ . . . sto A kai kn ∈ N pouikanopoioÔn ta ex c:

• Gia k�je a ∈ A isqÔei a < an−1 + kn

2n−1 .

• an−1 + kn−12n−1 ≤ an.

Apìdeixh tou epagwgikoÔ b matoc: 'Eqoume an ∈ A kai apì thn sunj kh 1up�rqei el�qistoc fusikìc kn+1 me thn idiìthta: gia k�je a ∈ A,

a < an +kn+1

2n.

Autì shmaÐnei ìti up�rqei an+1 me

an +kn+1 − 1

2n≤ an+1.

Isqurismìc 1: H (an) eÐnai akoloujÐa Cauchy.

2.10 Askhseic · 61

Pr�gmati, èqoume

an−1 +kn − 12n−1

≤ an < an−1 +kn

2n−1,

�ra

|an − an−1| < 12n−1

.

An loipìn n,m ∈ N kai n < m, tìte

|am − an| ≤ |am − am−1|+ |am−1 − am−2|+ · · ·+ |an+1 − an|<

12m−1

+1

2m−2+ . . . +

12n

<1

2n−1.

An ta n,m eÐnai arket� meg�la, autì gÐnetai ìso jèloume mikrì. Pio sugkekri-mèna, an mac d¸soun ε > 0, up�rqei n0 ∈ N t.w 1/2n0−1 < ε, opìte gia k�jen,m ≥ n0 èqoume |am − an| < ε. 2

AfoÔ to R∗ èqei thn idiìthta Cauchy, up�rqei o a∗ = lim an.Isqurismìc 2: O a∗ eÐnai to el�qisto �nw fr�gma tou A.(a) O a∗ eÐnai �nw fr�gma tou A: ac upojèsoume ìti up�rqei a ∈ A me a > a∗.MporoÔme na broÔme ε > 0 t.w a > a∗ + ε. 'Omwc,

a < an−1 +kn

2n−1≤ an +

12n−1

gia k�je n ∈ N. 'Ara,

a∗ + ε < an +1

2n−1

=⇒ a∗ + ε ≤ lim(

an +1

2n−1

)

=⇒ a∗ + ε ≤ a∗,

to opoÐo eÐnai �topo.(b) An a∗∗ eÐnai �nw fr�gma tou A, tìte a∗∗ ≥ a∗: èqoume a∗∗ ≥ an gia k�jen ∈ N (giatÐ?), �ra

a∗∗ ≥ lim an = a∗.

Apì ta (a) kai (b) eÐnai safèc ìti a∗ = sup A. 2

2.10 Ask seic



1. K�je fragmènh akoloujÐa sugklÐnei.

2. K�je sugklÐnousa akoloujÐa eÐnai fragmènh.


3. An (an) eÐnai mia akoloujÐa akeraÐwn arijm¸n, tìte h (an) sugklÐnei an kaimìno an eÐnai stajer telik�.

4. Up�rqei gnhsÐwc fjÐnousa akoloujÐa fusik¸n arijm¸n.

5. K�je sugklÐnousa akoloujÐa �rrhtwn arijm¸n sugklÐnei se �rrhto arijmì.

6. K�je pragmatikìc arijmìc eÐnai ìrio k�poiac akoloujÐac �rrhtwn arijm¸n.

7. An (an) eÐnai mia akoloujÐa jetik¸n pragmatik¸n arijm¸n, tìte an → 0 ankai mìno an 1

an→ +∞.

8. An an → a tìte h (an) eÐnai monìtonh.

9. 'Estw (an) aÔxousa akoloujÐa. An h (an) den eÐnai �nw fragmènh, tìtean → +∞.

10. An h (an) eÐnai fragmènh kai h (bn) sugklÐnei tìte h (anbn) sugklÐnei.

11. An h (|an|) sugklÐnei tìte kai h (an) sugklÐnei.

12. An (an) eÐnai mia akoloujÐa jetik¸n pragmatik¸n arijm¸n me an → a > 0,tìte

inf{an : n ∈ N} > 0.

13. An (an) eÐnai mia akoloujÐa jetik¸n pragmatik¸n arijm¸n me an → 0, tìteto sÔnolo A = {an : n ∈ N} èqei mègisto stoiqeÐo.

14. An an > 0 kai h (an) den eÐnai �nw fragmènh, tìte an → +∞.

15. an → +∞ an kai mìno an gia k�je M > 0 up�rqoun �peiroi ìroi thc (an)pou eÐnai megalÔteroi apì M .

16. H (an) den eÐnai �nw fragmènh an kai mìno an up�rqei upakoloujÐa (akn)thc (an) ¸ste akn → +∞.

17. K�je upakoloujÐa miac sugklÐnousac akoloujÐac sugklÐnei.

18. An mia akoloujÐa den èqei fjÐnousa upakoloujÐa tìte èqei mia gnhsÐwcaÔxousa upakoloujÐa.

19. An an 6→ a tìte up�rqoun b 6= a kai upakoloujÐa (akn) thc (an) ¸steakn → b.

20. Up�rqei fragmènh akoloujÐa pou den èqei sugklÐnousa upakoloujÐa.

21. An h (an) den eÐnai fragmènh, tìte den èqei fragmènh upakoloujÐa.

2.10 Askhseic · 63

22. 'Estw (an) aÔxousa akoloujÐa. K�je upakoloujÐa thc (an) eÐnai aÔxousa.

23. An h (an) eÐnai aÔxousa kai gia k�poia upakoloujÐa (akn) thc (an) èqoumeakn → a, tìte an → a.

24. An an → 0 tìte up�rqei upakoloujÐa (akn) thc (an) ¸ste n2akn

→ 0.

25. 'Estw (an) mia akoloujÐa. An sup{an : n ∈ N} = 1 kai an 6= 1 gia k�jen ∈ N, tìte up�rqei gnhsÐwc aÔxousa upakoloujÐa (akn) thc (an) ¸ste akn → 1.

26. An h (an) sugklÐnei kai an+2 = an gia k�je n ∈ N, tìte h (an) eÐnaistajer .


1. Exet�ste poièc apì tic parak�tw anisìthtec isqÔoun (a) telik�, (b) gia�peirouc to pl joc deÐktec:

n+(−1)n > 103, (−1)nn > 1000,1n

+(−1)n

n2< 10−10, (1+(−1)n)n < 2, n2+2n < 106.

2. 'Estw (an) akoloujÐa pragmatik¸n arijm¸n me limn→∞

an = 2. JewroÔme tasÔnola

A1 = {n ∈ N : an < 2.001}A2 = {n ∈ N : an > 2.003}A3 = {n ∈ N : an < 1.98}A4 = {n ∈ N : 1.991 < an < 2.01}A5 = {n ∈ N : 1.99997 < an < 2.0001}A6 = {n ∈ N : an ≤ 2}.

Gia k�je j = 1, . . . , 6 exet�ste an (a) to Aj eÐnai peperasmèno, (b) to N\Aj eÐnaipeperasmèno.

3. ApodeÐxte me ton orismì ìti oi parak�tw akoloujÐec sugklÐnoun sto 0:

an =n

n3 + n2 + 1, bn =

√n2 + 2−

√n2 + 1, cn =

{12n n = 1, 4, 7, 10, 13, . . .1

n2+1 alli¸c.

4. ApodeÐxte me ton orismì ìti

an =n2 − n

n2 + n→ 1.

5. 'Estw (an) akoloujÐa pragmatik¸n arijm¸n. An limn→∞

an = a > 0, deÐxte ìtian > 0 telik�.

6. 'Estw (an), (bn) dÔo akoloujÐec me an → a kai bn → b.

(a) An an ≤ bn gia k�je n ∈ N, deÐxte ìti a ≤ b.

(b) An an < bn gia k�je n ∈ N, mporoÔme na sumper�noume ìti a < b?

(g) An m ≤ an ≤ M gia k�je n ∈ N, deÐxte ìti m ≤ a ≤ M .

7. 'Estw (an) akoloujÐa pragmatik¸n arijm¸n.

(a) DeÐxte ìti an → 0 an kai mìno an |an| → 0.

(b) DeÐxte ìti an an → a 6= 0 tìte |an| → |a|. IsqÔei to antÐstrofo?

(g) 'Estw k ≥ 2. DeÐxte ìti an an → a tìte k√|an| → k

√|a|.

8. (a) 'Estw µ > 1 kai an > 0 gia k�je n ∈ N. An an+1 ≥ µan gia k�je n,deÐxte ìti an → +∞.

(b) 'Estw 0 < µ < 1 kai (an) akoloujÐa me thn idiìthta |an+1| ≤ µ|an| gia k�jen. DeÐxte ìti an → 0.

(g) 'Estw an > 0 gia k�je n, kai an+1an

→ ` > 1. DeÐxte ìti an → +∞.

(d) 'Estw an 6= 0 gia k�je n, kai∣∣∣∣an+1

an

∣∣∣∣ → ` < 1. DeÐxte ìti an → 0.

9. (a) 'Estw a > 0. DeÐxte ìti n√

a → 1.

(b) 'Estw (an) akoloujÐa jetik¸n pragmatik¸n arijm¸n. An an → a > 0 tìten√

an → 1. Ti mporeÐte na peÐte an an → 0?

(g) DeÐxte ìti n√

n → 1.

10. (a) 'Estw a ∈ R me |a| < 1. DeÐxte ìti h akoloujÐa bn = an sugklÐnei sto0.

(b) Gia poièc timèc tou x ∈ R sugklÐnei h akoloujÐa(

1−x2

1+x2

)n

?

11. UpologÐste ta ìria twn parak�tw akolouji¸n:

(a) an = n3+5n2+22n3+9 .

(b) bn = 1n2 + 1

(n+1)2 + · · ·+ 1(2n)2 .

(g) cn = 1√n2+1

+ 1√n2+2

+ · · ·+ 1√n2+n

.

(d) dn = n2n .

(e) en = 2n+n3n−n .

(st) fn = n4n .

(z) gn = n5

n! .

(h) hn = nn

n! .

(j) xn = 1+22+33+···+nn

nn .

12. 'Estw A mh kenì kai �nw fragmèno uposÔnolo tou R. An a = sup A, deÐxteìti up�rqei akoloujÐa (an) stoiqeÐwn tou A me lim

n→∞an = a.

2.10 Askhseic · 65

An, epiplèon, to supA den eÐnai stoiqeÐo tou A, deÐxte ìti h parap�nw ako-loujÐa mporeÐ na epilegeÐ ¸ste na eÐnai gnhsÐwc aÔxousa.

13. DeÐxte ìti k�je pragmatikìc arijmìc eÐnai ìrio gnhsÐwc aÔxousac akoloujÐ-ac rht¸n arijm¸n, kaj¸c epÐshc kai ìrio gnhsÐwc aÔxousac akoloujÐac �rrhtwnarijm¸n.

14. OrÐzoume mia akoloujÐa (an) me a1 =√

2 kai an+1 =√

2 +√

an, n ∈ N.Exet�ste an sugklÐnei.

15. OrÐzoume mia akoloujÐa (an) me a1 = 1 kai an+1 = 2an+1an+1 , n ∈ N. Exet�ste

an sugklÐnei.

16. 'Estw a > 0. JewroÔme tuqìn x1 > 0 kai gia k�je n ∈ N orÐzoume

xn+1 =12

(xn +

a

xn

).

DeÐxte ìti h (xn), toul�qiston apì ton deÔtero ìro thc kai pèra, eÐnai fjÐnousakai k�tw fragmènh apì ton

√a. BreÐte to lim

n→∞xn.

17. 'Estw (an) akoloujÐa me an → a. OrÐzoume mia deÔterh akoloujÐa (bn)jètontac

bn =a1 + · · ·+ an

n.

DeÐxte ìti bn → a.

18. 'Estw (an) akoloujÐa jetik¸n arijm¸n. JewroÔme to sÔnolo A = {an :n ∈ N}. An inf A = 0, deÐxte ìti h (an) èqei fjÐnousa upakoloujÐa pou sugklÐneisto 0.

19. 'Estw (an) mia akoloujÐa. DeÐxte ìti an → a an kai mìno an oi upakoloujÐec(a2k) kai (a2k−1) sugklÐnoun sto a.

20. 'Estw (an) mia akoloujÐa. Upojètoume ìti oi upakoloujÐec (a2k), (a2k−1)kai (a3k) sugklÐnoun. DeÐxte ìti:(a) lim

k→∞a2k = lim

k→∞a2k−1 = lim

k→∞a3k.

(b) H (an) sugklÐnei.

21. 'Estw (an) mia akoloujÐa. Upojètoume ìti a2n ≤ a2n+2 ≤ a2n+1 ≤ a2n−1

gia k�je n ∈ N kai ìti limn→∞

(a2n−1−a2n) = 0. Tìte h (an) sugklÐnei se k�poionpragmatikì arijmì a pou ikanopoieÐ thn a2n ≤ a ≤ a2n−1 gia k�je n ∈ N.

22. JewroÔme èna kleistì di�sthma [a, b] kai mia akoloujÐa (xn) pragmatik¸narijm¸n pou an koun sto [a, b].(a) DeÐxte ìti up�rqei akoloujÐa ([ak, bk]) kleist¸n diasthm�twn pou ikanopoieÐta ex c:

(i) Gia k�je k ∈ N isqÔei [ak+1, bk+1] ⊂ [ak, bk].

(ii) Gia k�je k ∈ N isqÔei bk − ak = (b− a)/2k−1.

(iii) Gia k�je k ∈ N up�rqoun �peiroi ìroi thc (xn) sto [ak, bk].

(b) Qrhsimopoi¸ntac thn arq tou kibwtismoÔ kai to (a) deÐxte ìti h (xn) èqeiupakoloujÐa (xkn

) pou sugklÐnei se k�poio x ∈ [a, b].

23. DeÐxte ìti h akoloujÐa (an) den sugklÐnei ston pragmatikì arijmì a, an kaimìno an up�rqoun ε > 0 kai upakoloujÐa (akn) thc (an) ¸ste |akn − a| ≥ ε giak�je n ∈ N.

24. 'Estw (an) akoloujÐa pragmatik¸n arijm¸n kai èstw a ∈ R. DeÐxte ìtian → a an kai mìno an k�je upakoloujÐa thc (an) èqei upakoloujÐa pou sugklÐneisto a.

25. Qrhsimopoi¸ntac thn anisìthta

1n + 1

+1

n + 2+ · · ·+ 1

2n≥ 1

2,

deÐxte ìti h akoloujÐa an = 1 + 12 + · · ·+ 1

n den eÐnai akoloujÐa Cauchy. Sumpe-r�nate ìti an → +∞.

26. 'Estw 0 < µ < 1 kai akoloujÐa (an) gia thn opoÐa isqÔei

|an+1 − an| ≤ µ|an − an−1|, n ≥ 2.

DeÐxte ìti h (an) eÐnai akoloujÐa Cauchy.

27. OrÐzoume a1 = a, a2 = b kai an+1 = an+an−12 , n ≥ 2. Exet�ste an h (an)

eÐnai akoloujÐa Cauchy.

28. 'Estw (an) mia akoloujÐa. OrÐzoume

bn = sup{|an+k − an| : k ∈ N}.

DeÐxte ìti h (an) sugklÐnei an kai mìno an bn → 0.

2.10 Askhseic · 67

G. Ask seic

1. (a) 'Estw a1, a2, . . . , ak > 0. DeÐxte ìti

bn := n√

an1 + an

2 + · · ·+ ank → max{a1, a2, . . . , ak}.

(b) UpologÐste to ìrio thc akoloujÐac

xn =1n

n√

1n + 2n + · · ·+ nn.

2. UpologÐste ta ìria twn parak�tw akolouji¸n:

an =(

1 +1

n− 1

)n−1

, bn =(

1 +2n

)n

, cn =(

1− 1n

)n

kaidn =

(1− 1

n2

)n

, en =(

1 +23n

)n

.

3. 'Estw 0 < a1 < b1. OrÐzoume anadromik� dÔo akoloujÐec jètontac

an+1 =√

anbn kai bn+1 =an + bn

2.

(a) DeÐxte ìti h (an) eÐnai aÔxousa kai h (bn) fjÐnousa.(b) DeÐxte ìti oi (an), (bn) sugklÐnoun kai èqoun to Ðdio ìrio.

4. Epilègoume x1 = a, x2 = b kai jètoume

xn+2 =xn

3+

2xn+1

3.

DeÐxte ìti h (xn) sugklÐnei kai breÐte to ìriì thc. [Upìdeixh: Jewr ste thnyn = xn+1 − xn kai breÐte anadromikì tÔpo gia thn (yn).]

5. (L mma tou Stoltz) 'Estw (an) akoloujÐa pragmatik¸n arijm¸n kai èstw (bn)gnhsÐwc aÔxousa akoloujÐa pragmatik¸n arijm¸n me lim

n→∞bn = +∞. DeÐxte ìti

anlim

n→∞an+1 − an

bn+1 − bn= λ,

ìpou λ ∈ R λ = +∞, tìtelim

n→∞an

bn= λ.

6. 'Estw (an) akoloujÐa jetik¸n ìrwn me an → a > 0. DeÐxte ìti

bn :=n

1a1

+ · · ·+ 1an

→ a kai γn := n√

a1 · · · an → a.

7. 'Estw (an) akoloujÐa me limn→∞

(an+1 − an) = a. DeÐxte ìti

an

n→ a.


8. 'Estw (an) aÔxousa akoloujÐa me thn idiìthta

bn :=a1 + · · ·+ an

n→ a.

DeÐxte ìti an → a.

9. OrÐzoume mia akoloujÐa (an) me a1 > 0 kai

an+1 = 1 +2

1 + an.

DeÐxte ìti oi upakoloujÐec (a2k) kai (a2k−1) eÐnai monìtonec kai fragmènec. BreÐ-te, an up�rqei, to lim

n→∞an.

Kef�laio 3

SuneqeÐc sunart seic

3.1 Sunart seic

'Estw X kai Y dÔo mh ken� sÔnola. Me ton ìro sun�rthsh apì to X stoY ennooÔme mia antistoÐqish pou stèlnei k�je stoiqeÐo x tou X se èna kaimonadikì stoiqeÐo y tou Y . MporoÔme na kwdikopoi soume thn plhroforÐa ìtito x apeikonÐzetai sto y qrhsimopoi¸ntac to diatetagmèno zeÔgoc (x, y): to pr¸tostoiqeÐo x tou zeÔgouc eÐnai sto X kai to deÔtero eÐnai to stoiqeÐo tou Y stoopoÐo antistoiqÐzoume to x. OdhgoÔmaste ètsi ston ex c orismì:

Orismìc 3.1.1. 'Estw X kai Y dÔo mh ken� sÔnola. JewroÔme to kartesianìginìmeno twn X kai Y :

X × Y = {(x, y) : x ∈ X, y ∈ Y }.

Sun�rthsh f apì to X sto Y lègetai k�je uposÔnolo f tou X ×Y to opoÐoikanopoieÐ ta ex c:

(i) Gia k�je x ∈ X up�rqei y ∈ Y ¸ste (x, y) ∈ f . H sunj kh aut perigr�feito gegonìc ìti apaitoÔme k�je x ∈ X na apeikonÐzetai se k�poio y ∈ Y .

(ii) An (x, y1) ∈ f kai (x, y2) ∈ f , tìte y1 = y2. H sunj kh aut perigr�fei togegonìc ìti apaitoÔme k�je x ∈ X na èqei monos manta orismènh eikìnay ∈ Y .

Gr�fontac f : X → Y ennooÔme ìti f eÐnai mia sun�rthsh apì to X sto Y .SumfwnoÔme epÐshc na gr�foume y = f(x) gia thn eikìna tou x mèsw thc f .Dhlad , y = f(x) ⇐⇒ (x, y) ∈ f .

'Estw f : X → Y mia sun�rthsh. Lème ìti to X eÐnai to pedÐo orismoÔthc f kai to Y eÐnai to pedÐo tim¸n thc f . To sÔnolo tim¸n ( eikìna)thc f eÐnai to sÔnolo

f(X) = {y ∈ Y : up�rqei x ∈ X ¸ste f(x) = y} = {f(x) : x ∈ X}.

70 · Suneqeic sunarthseic

ParadeÐgmata 3.1.2. (a) 'Estw c ∈ R. H sun�rthsh f : R→ R me f(x) = c

gia k�je x ∈ R lègetai stajer sun�rthsh. To sÔnolo tim¸n thc f eÐnaito monosÔnolo f(X) = {c}.

(b) H sun�rthsh f : R → R me f(x) = x. To sÔnolo tim¸n thc f eÐnai tosÔnolo f(X) = R.

(g) H sun�rthsh f : R → R me f(x) = x2. To sÔnolo tim¸n thc f eÐnai tosÔnolo f(X) = [0, +∞).

(d) H sun�rthsh f : R → R me f(x) = 1 an x ∈ Q kai f(x) = 0 an x /∈ Q.To sÔnolo tim¸n thc f eÐnai to sÔnolo f(X) = {0, 1}.

(e) H sun�rthsh f : R → R me f(x) = 1q an x 6= 0 rhtìc o opoÐoc gr�fetai

sth morf x = pq ìpou p ∈ Z, q ∈ N, MKD(p, q) = 1, kai f(x) = 0 an x /∈ Q

x = 0. To sÔnolo tim¸n thc f eÐnai to sÔnolo f(X) = {0} ∪ {1/n : n ∈ N}.

Orismìc 3.1.3. 'Estw f : X → Y mia sun�rthsh. H f lègetai epÐ anf(X) = Y , dhlad an gia k�je y ∈ Y up�rqei x ∈ X ¸ste f(x) = y.

H sun�rthsh f lègetai 1-1 an apeikonÐzei diaforetik� stoiqeÐa tou X sediaforetik� stoiqeÐa tou Y . Dhlad , an gia k�je x1, x2 ∈ X me x1 6= x2 èqoumef(x1) 6= f(x2). IsodÔnama, gia na elègxoume ìti h f eÐnai 1-1 prèpei na deÐxoumeìti an x1, x2 ∈ X kai f(x1) = f(x2), tìte x1 = x2.

Orismìc 3.1.4 (sÔnjesh sunart sewn). 'Estw f : X → Y kai g :W → Z dÔo sunart seic. Upojètoume ìti f(X) ⊆ W , dhlad , h eikìna thcf perièqetai sto pedÐo orismoÔ thc g. Tìte, an x ∈ X èqoume f(x) ∈ W kaiorÐzetai h eikìna g(f(x)) tou f(x) mèsw thc g. MporoÔme loipìn na orÐsoumemia sun�rthsh g ◦ f : X → Z, jètontac

(g ◦ f)(x) = g(f(x)) (x ∈ X).

H sun�rthsh g ◦ f lègetai sÔnjesh thc g me thn f .

Orismìc 3.1.5 (eikìna kai antÐstrofh eikìna). 'Estw f : X → Y

mia sun�rthsh.

(a) Gia k�je A ⊆ X, h eikìna tou A mèsw thc f eÐnai to sÔnolo

f(A) = {y ∈ Y : up�rqei x ∈ A ¸ste f(x) = y} = {f(x) : x ∈ A}.

(b) Gia k�je B ⊆ Y , h antÐstrofh eikìna tou B mèsw thc f eÐnai to sÔnolo

f−1(B) = {x ∈ X : f(x) ∈ B}.

Prìtash 3.1.6. 'Estw f : X → Y mia sun�rthsh. IsqÔoun ta ex c:

(i) An A1 ⊆ A2 ⊆ X, tìte f(A1) ⊆ f(A2).

(ii) An A1, A2 ⊆ X, tìte f(A1 ∪A2) = f(A1) ∪ f(A2).

(iii) An A1, A2 ⊆ X, tìte f(A1 ∩ A2) ⊆ f(A1) ∩ f(A2). O egkleismìc mporeÐna eÐnai gn sioc. IsqÔei ìmwc p�nta isìthta an h f eÐnai 1-1.

3.1 Sunarthseic · 71

(iv) An B1 ⊆ B2 ⊆ Y tìte f−1(B1) ⊆ f−1(B2).

(v) An B1, B2 ⊆ Y , tìte f−1(B1 ∪B2) = f−1(B1) ∪ f−1(B2).

(vi) An B1, B2 ⊆ Y , tìte f−1(B1 ∩B2) = f−1(B1) ∩ f−1(B2).

(vii) An B ⊆ Y tìte f−1(Y \B) = X \ f−1(B).

(viii) An A ⊆ X tìte A ⊆ f−1(f(A)). O egkleismìc mporeÐ na eÐnai gn sioc.IsqÔei ìmwc p�nta isìthta an h f eÐnai 1-1.

(ix) An B ⊆ Y tìte f(f−1(B)) ⊆ B. O egkleismìc mporeÐ na eÐnai gn sioc.IsqÔei ìmwc p�nta isìthta an h f eÐnai epÐ.

Orismìc 3.1.7 (antÐstrofh sun�rthsh). 'Estw f : X → Y mia 1-1sun�rthsh. MporoÔme na jewr soume th sun�rthsh f san sun�rthsh apì toX sto f(X) (h f paÐrnei timèc sto sÔnolo f(X)). H f : X → f(X) eÐnai 1-1kai epÐ. Sunep¸c, gia k�je y ∈ f(X) up�rqei x ∈ X ¸ste f(x) = y, kai autìto x ∈ X eÐnai monadikì afoÔ h f eÐnai 1-1. MporoÔme loipìn na orÐsoume miasun�rthsh f−1 : f(X) → X, wc ex c:

f−1(y) = x, ìpou x eÐnai to monadikì x ∈ X gia to opoÐo f(x) = y.

Me �lla lìgia,f−1(y) = x ⇐⇒ f(x) = y.

H f−1 eÐnai kal� orismènh sun�rthsh apì to f(X) sto X, h antÐstrofhsun�rthsh thc f .

Prìtash 3.1.8. 'Estw f : X → Y mia 1-1 sun�rthsh. Oi f−1 ◦ f : X → X

kai f ◦ f−1 : f(X) → f(X) orÐzontai kal� kai ikanopoioÔn tic:

(a) (f−1 ◦ f)(x) = x gia k�je x ∈ X.

(b) (f ◦ f−1)(y) = y gia k�je y ∈ f(X).

Orismìc 3.1.9 (pr�xeic kai di�taxh). 'Estw A èna mh kenì sÔnolo kaièstw f : A → R kai g : A → R dÔo sunart seic me pedÐo tim¸n to R. Tìte,

(i) H sun�rthsh f + g : A → R orÐzetai wc ex c: (f + g)(x) = f(x) + g(x)gia k�je x ∈ A.

(ii) H sun�rthsh f · g : A → R orÐzetai wc ex c: (f · g)(x) = f(x) · g(x) giak�je x ∈ A.

(iii) Gia k�je t ∈ R orÐzetai h sun�rthsh tf : A → R me (tf)(x) = tf(x) giak�je x ∈ A.

(iv) An g(x) 6= 0 gia k�je x ∈ A, tìte orÐzetai hf

g: A → R me

(f

g

)(x) =

f(x)g(x)

gia k�je x ∈ A.

Lème ìti f ≤ g an f(x) ≤ g(x) gia k�je x ∈ A.

Orismìc 3.1.10 (monìtonec sunart seic). 'Estw A èna mh kenì upo-sÔnolo tou R kai èstw f : A → R mia sun�rthsh. Lème ìti:

(i) H f eÐnai aÔxousa an gia k�je x, y ∈ A me x < y isqÔei f(x) ≤ f(y).

(ii) H f eÐnai gnhsÐwc aÔxousa an gia k�je x, y ∈ A me x < y isqÔeif(x) < f(y).

(iii) H f eÐnai fjÐnousa an gia k�je x, y ∈ A me x < y isqÔei f(x) ≥ f(y).

(iv) H f eÐnai gnhsÐwc fjÐnousa an gia k�je x, y ∈ A me x < y isqÔeif(x) > f(y).

(v) H f eÐnai monìtonh an eÐnai aÔxousa fjÐnousa.

(vi) H f eÐnai gnhsÐwc monìtonh an eÐnai gnhsÐwc aÔxousa gnhsÐwc fjÐ-nousa.

Orismìc 3.1.11 (fragmènh sun�rthsh). 'Estw A èna mh kenì sÔnolokai èstw f : A → R mia sun�rthsh. Lème ìti:

(i) H f eÐnai �nw fragmènh an up�rqei M ∈ R ¸ste gia k�je x ∈ A naisqÔei f(x) ≤ M .

(ii) H f eÐnai k�tw fragmènh an up�rqei m ∈ R ¸ste gia k�je x ∈ A naisqÔei f(x) ≥ m.

(iii) H f eÐnai fragmènh an eÐnai �nw kai k�tw fragmènh. IsodÔnama, anup�rqei M > 0 ¸ste gia k�je x ∈ A na isqÔei |f(x)| ≤ M .

3.1aþ Ask seic

1. 'Estw a, b ∈ R me a < b. DeÐxte ìti h apeikìnish f : [0, 1] → [a, b] : x →a + (b− a)x eÐnai 1-1 kai epÐ.

2. 'Estw f, g : [0, 1] → [0, 1] me f(x) = 1−x1+x kai g(t) = 4t(1− t).

(a) Na breÐte tic f ◦ g kai g ◦ f .(b) Na deÐxete ìti orÐzetai h f−1 all� den orÐzetai h g−1.

3. 'Estw g : X → Y, f : Y → Z dÔo sunart seic pou eÐnai 1-1 kai epÐ. DeÐxte ìtiorÐzetai h antÐstrofh sun�rthsh (f ◦g)−1 thc f ◦g kai ìti (f ◦g)−1 = g−1 ◦f−1.

4. 'Estw g : X → Y, f : Y → Z dÔo sunart seic. DeÐxte ìti

(a) an h f ◦ g eÐnai epÐ tìte kai h f eÐnai epÐ.

(b) an h f ◦ g eÐnai 1-1 tìte kai h g eÐnai 1-1.


IsqÔoun ta antÐstrofa twn (a) kai (b)?

5. 'Estw f : X → X mia sun�rthsh. Upojètoume ìti up�rqoun sunart seicg : X → X kai h : X → X ¸ste f ◦ g = IdX kai h ◦ f = IdX . DeÐxte ìti h = g.

6. An A ⊆ R, sumbolÐzoume me χA : R → R thn qarakthristik su-

n�rthsh tou A pou orÐzetai apì thn χA(x) ={

1 an x ∈ A

0 an x /∈ A. ApodeÐxte

ìti

(a) χA∩B = χA.χB (eidikìtera χA = χ2A),

(b) χA∪B = χA + χB − χA.χB ,

(g) χR\A = 1− χA,

(d) A ⊆ B ⇐⇒ χA ≤ χB kai

(e) An f : R → R eÐnai mia sun�rthsh me f2 = f , tìte up�rqei A ⊆ R ¸stef = χA.

7. 'Estw f(x) = 11+x .

(a) Na brejeÐ to pedÐo orismoÔ thc f .

(b) Na brejeÐ h f ◦ f .

(g) Na brejoÔn ta f( 1x ), f(cx), f(x + y), f(x) + f(y).

(d) Gia poi� c ∈ R up�rqei x ∈ R ¸ste f(cx) = f(x)?

(e) Gia poi� c ∈ R h sqèsh f(cx) = f(x) ikanopoieÐtai gia dÔo diaforetikèctimèc tou x ∈ R?

8. Mia sun�rthsh g : R → R lègetai �rtia an g(x) = g(−x) gia k�je x ∈ Rkai peritt an g(x) = −g(−x) gia k�je x ∈ R. DeÐxte ìti k�je sun�rthshf : R → R gr�fetai wc �jroisma f = fa + fp ìpou fa �rtia kai fp peritt , kaiìti aut h anapar�stash eÐnai monadik .

9. Mia sun�rthsh f : R→ R lègetai periodik (me perÐodo a) an up�rqeia 6= 0 sto R ¸ste f(x + a) = f(x) gia k�je x ∈ R.(a) DeÐxte ìti h sun�rthsh f : R→ R pou orÐzetai apì thn f(x) = [x] den eÐnaiperiodik .(b) Exet�ste an h sun�rthsh f : R → R pou orÐzetai apì thn f(x) = x − [x]eÐnai periodik .

10. 'Estw n ∈ N.(a) DeÐxte ìti h sun�rthsh

f(x) = [x] +[x +

1n

]+ · · ·+

[x +

n− 1n

]− [nx]

eÐnai periodik me perÐodo 1/n. Dhlad , f(x + 1

n

)= f(x) gia k�je x ∈ R.

(b) UpologÐste thn tim f(x) ìtan 0 ≤ x < 1/n.

(g) DeÐxte thn tautìthta

[nx] = [x] +[x +

1n

]+ · · ·+

[x +

n− 1n

]

gia k�je x ∈ R kai k�je n ∈ N.

11. An mia sun�rthsh f eÐnai gnhsÐwc aÔxousa sta diast mata I1 kai I2, eÐnaial jeia ìti eÐnai gnhsÐwc aÔxousa sto I1 ∪ I2?

12. 'Estw f(x) ={

x + 1 an x ≤ 1x2 + 1 an x ≥ 1

. 'Exet�ste an eÐnai monìtonh kai breÐte

thn f−1 (an aut orÐzetai).

13. 'Estw f(x) = x+1. Na brejeÐ mia sun�rthsh g : R→ R ¸ste g ◦ f = f ◦ g.EÐnai h g monadik ?

14. ApodeÐxte ìti h sun�rthsh f(x) = x|x|+1 eÐnai gnhsÐwc aÔxousa kai fragmènh

sto R. Poiì eÐnai to sÔnolo tim¸n f(R)?

15. 'Estw f : R→ R mia sun�rthsh me f(x+y) = f(x)+f(y) gia k�je x, y ∈ R.ApodeÐxte ìti

(a) f(0) = 0 kai f(−x) = −f(x) gia k�je x ∈ R.

(b) Gia k�je n ∈ N kai x1, x2, . . . , xn ∈ R isqÔei

f(x1 + x2 + · · ·+ xn) = f(x1) + f(x2) + · · ·+ f(xn).

(g) f( 1n ) = f(1)

n gia k�je n ∈ N.

(d) Up�rqei λ ∈ R ¸ste f(q) = λq gia k�je q ∈ Q.

16. 'Estw f : R→ R mia sun�rthsh me f(y)−f(x) ≤ (y−x)2 gia k�je x, y ∈ R.ApodeÐxte ìti h f eÐnai stajer .

[Upìdeixh: An |f(b) − f(a)| = δ > 0, diairèste to di�sthma [a, b] se n Ðsaupodiast mata, ìpou n arket� meg�loc fusikìc arijmìc.]

3.2 Suneqeia sunarthshc · 75

3.2 Sunèqeia sun�rthshc

3.2aþ Orismìc thc sunèqeiac

Orismìc 3.2.1. 'Estw A èna mh kenì uposÔnolo tou R, èstw f : A → R kaièstw x0 ∈ A. Lème ìti h f eÐnai suneq c sto x0 an: gia k�je ε > 0 up�rqeiδ > 0 ¸ste:

an x ∈ A kai |x− x0| < δ, tìte |f(x)− f(x0)| < ε.

Lème ìti h f eÐnai suneq c sto A an eÐnai suneq c se k�je x0 ∈ A.

Parathr seic 3.2.2. (a) To dojèn ε > 0 kajorÐzei mia perioq (f(x0) −ε, f(x0) + ε) thc tim c f(x0). Autì pou zht�me eÐnai na mporoÔme na broÔmemia perioq (x0 − δ, x0 + δ) tou x0 ¸ste k�je x ∈ A pou an kei se aut n thnperioq tou x0 na apeikonÐzetai sto (f(x0) − ε, f(x0) + ε). Dhlad , na isqÔeif((x0 − δ, x0 + δ) ∩A) ⊆ (f(x0)− ε, f(x0) + ε).

An to parap�nw isqÔei gia k�je ε > 0, tìte lème ìti h f eÐnai suneq c stox0.

(b) Apì ton orismì eÐnai fanerì ìti exet�zoume th sunèqeia mìno sta shmeÐa toupedÐou orismoÔ thc f .

ParadeÐgmata 3.2.3. (a) f : R→ R me f(x) = c gia k�je x ∈ R. JadeÐxoume ìti h f eÐnai suneq c se k�je x0 ∈ R. 'Estw ε > 0. Zht�me δ > 0 ¸ste:an x ∈ R kai |x−x0| < δ, tìte |f(x)−f(x0)| < ε. 'Omwc, gia k�je x ∈ R èqoume

|f(x)− f(x0)| = |c− c| = 0 < ε.

Dhlad , mporoÔme na epilèxoume opoiod pote δ > 0 (gia par�deigma, δ = 100).

(b) f : R→ R me f(x) = x gia k�je x ∈ R. Ja deÐxoume ìti h f eÐnai suneq c sek�je x0 ∈ R. 'Estw ε > 0. Zht�me δ > 0 ¸ste: an x ∈ R kai |x− x0| < δ, tìte|f(x) − f(x0)| < ε. AfoÔ |f(x) − f(x0)| = |x − x0|, arkeÐ na epilèxoume δ = ε.Tìte,

|x− x0| < δ =⇒ |f(x)− f(x0)| = |x− x0| < δ = ε.

Parathr ste ìti, se autì to par�deigma, to δ exart�tai apì to ε all� den exar-t�tai apì to x0.

(g) f : R→ R me f(x) = 2x2 − 1 gia k�je x ∈ R. Ja deÐxoume ìti h f eÐnaisuneq c se k�je x0 ∈ R. 'Estw ε > 0. Zht�me δ > 0 ¸ste: an x ∈ R kai|x− x0| < δ, tìte |f(x)− f(x0)| < ε.

Parathr ste ìti, gia k�je x ∈ R,|f(x)− f(x0)| = |(2x2 − 1)− (2x2

0 − 1)| = |2x2 − 2x20| = 2|x + x0| · |x− x0|.

Zht�me loipìn δ > 0 ¸ste: an |x−x0| < δ, tìte 2|x+x0|·|x−x0| < ε. Dedomènouìti emeÐc eÐmaste pou ja k�noume thn epilog tou δ, mporoÔme na upojèsoumeapì thn arq ìti to δ ja eÐnai mikrìtero apì 1. Tìte, an |x− x0| < δ ja èqoume|x− x0| < 1, kai sunep¸c,

|x + x0| ≤ |x− x0 + 2x0| ≤ |x− x0|+ 2|x0| < 1 + 2|x0|.

An, epiplèon, δ < ε2(2|x0|+1) , tìte, gia k�je x ∈ R me |x− x0| < δ ja èqoume

|f(x)− f(x0)| = 2|x + x0| · |x− x0| ≤ 2(2|x0|+ 1) |x− x0| < 2(2|x0|+ 1)δ < ε.

Dhlad , an epilèxoume

0 < δ < min{

1,ε

2(2|x0|+ 1)

},

èqoume|x− x0| < δ =⇒ |f(x)− f(x0)| < ε.

Parathr ste ìti to δ pou epilèxame exart�tai apì to dojèn ε all� kai apì toshmeÐo x0 sto opoÐo exet�zoume th sunèqeia thc f .

3.2bþ H �rnhsh tou orismoÔ

'Estw f : A → R kai èstw x0 ∈ A. Upojètoume ìti h f den eÐnai suneq c sto x0.Me b�sh th suz thsh pou ègine met� ton orismì thc sunèqeiac, autì shmaÐnei ìtiup�rqei k�poio ε me thn ex c idiìthta: an jewr soume opoiod pote δ > 0 kai thnantÐstoiqh perioq (x0−δ, x0+δ) tou x0, tìte den isqÔei f((x0−δ, x0+δ)∩A) ⊆(f(x0)− ε, f(x0)+ ε). Me �lla lìgia, up�rqei k�poio x ∈ A to opoÐo an kei sto(x0 − δ, x0 + δ) all� den ikanopoieÐ thn |f(x)− f(x0)| < ε. IsodÔnama,

Gia k�je δ > 0 up�rqei x ∈ A me |x− x0| < δ kai |f(x)− f(x0)| ≥ ε.

Katal goume loipìn sto ex c:

H f : A → R eÐnai asuneq c sto x0 ∈ A an kai mìno an up�rqeiε > 0 ¸ste: gia k�je δ > 0 up�rqei x ∈ A me |x − x0| < δ kai|f(x)− f(x0)| ≥ ε.

Me lìgia, ja lègame ìti h f eÐnai asuneq c sto x0 an {osod pote kont� stox0 up�rqei x ∈ A ¸ste oi timèc f(x) kai f(x0) na apèqoun arket�}.

Par�deigma 3.2.4. H sun�rthsh tou Dirichlet, f(x) =

1 x ∈ Q

0 x /∈ Q,

eÐnai asuneq c se k�je x0 ∈ R. Epilègoume ε = 12 > 0 kai ja deÐxoume ìti: gia

k�je δ > 0 up�rqei x ∈ R me |x−x0| < δ all� |f(x)− f(x0)| ≥ 12 . Pr�gmati, an

o x0 eÐnai rhtìc, parathroÔme ìti sto (x0−δ, x0 +δ) mporoÔme na broÔme �rrhtoα. Apì ton orismì thc f èqoume

|f(α)− f(x0)| = |0− 1| = 1 ≥ 12.

An o x0 eÐnai �rrhtoc, parathroÔme ìti sto (x0 − δ, x0 + δ) mporoÔme na broÔmerhtì q. Apì ton orismì thc f èqoume

|f(q)− f(x0)| = |1− 0| = 1 ≥ 12.

3.2gþ Arq thc metafor�c

'Estw f : A → R kai èstw x0 ∈ A. H arq thc metafor�c dÐnei ènanqarakthrismì thc sunèqeiac thc f sto x0 mèsw akolouji¸n.

Je¸rhma 3.2.5 (arq thc metafor�c). H f : A → R eÐnai suneq c stox0 ∈ A an kai mìno an: gia k�je akoloujÐa (xn) shmeÐwn tou A me xn → x0, hakoloujÐa (f(xn)) sugklÐnei sto f(x0).

Apìdeixh. Upojètoume pr¸ta ìti h f eÐnai suneq c sto x0. 'Estw xn ∈ A mexn → x0. Ja deÐxoume ìti f(xn) → f(x0): 'Estw ε > 0. AfoÔ h f eÐnai suneq csto x0, up�rqei δ > 0 ¸ste: an x ∈ A kai |x− x0| < δ, tìte |f(x)− f(x0)| < ε

(autìc eÐnai akrib¸c o orismìc thc sunèqeiac thc f sto x0).'Eqoume upojèsei ìti xn → x0. 'Ara, gi� autì to δ > 0 mporoÔme na broÔme

n0 ∈ N ¸ste: an n ≥ n0 tìte |xn − x0| < δ (autìc eÐnai akrib¸c o orismìc thcsÔgklishc thc (xn) sto x0).

Sundu�zontac ta parap�nw èqoume: an n ≥ n0, tìte |xn − x0| < δ �ra

|f(xn)− f(x0)| < ε.

AfoÔ to ε > 0 tan tuqìn, f(xn) → f(x0).

Gia thn antÐstrofh kateÔjunsh ja doulèyoume me apagwg se �topo. Upo-jètoume ìti gia k�je akoloujÐa (xn) shmeÐwn tou A me xn → x0, h akoloujÐa(f(xn)) sugklÐnei sto f(x0). Upojètoume epÐshc ìti h f den eÐnai suneq c stox0 kai ja katal xoume se �topo.

AfoÔ h f den eÐnai suneq c sto x0, up�rqei k�poio ε > 0 me thn ex c idiìthta:

(∗) Gia k�je δ > 0 up�rqei x ∈ A to opoÐo ikanopoieÐ thn |x−x0| < δ

all� |f(x)− f(x0)| ≥ ε.

QrhsimopoioÔme thn (∗) diadoqik� me δ = 1, 12 , . . . , 1

n , . . .. Gia k�je n ∈ Nèqoume 1/n > 0 kai apì thn (∗) brÐskoume xn ∈ A me |xn − x0| < 1/n kai|f(xn) − f(x0)| ≥ ε. Apì to krit rio parembol c eÐnai fanerì ìti xn → x0 kaiapì thn upìjesh pou k�name prèpei h akoloujÐa (f(xn)) na sugklÐnei sto f(x0).Autì ìmwc eÐnai adÔnato afoÔ |f(xn)− f(x0)| ≥ ε gia k�je n ∈ N. 2

Parat rhsh 3.2.6. H arq thc metafor�c mporeÐ na qrhsimopoihjeÐ me dÔodiaforetikoÔc trìpouc:

(i) gia na deÐxoume ìti h f eÐnai suneq c sto x0 arkeÐ na deÐxoume ìti {xn →x0 =⇒ f(xn) → f(x0)}.

(ii) gia na deÐxoume ìti h f den eÐnai suneq c sto x0 arkeÐ na broÔme mia ako-loujÐa xn → x0 (sto A) ¸ste limn f(xn) 6= f(x0). PolÔ suqn�, exasfa-lÐzoume thn asunèqeia thc f sto x0 brÐskontac dÔo akoloujÐec xn → x0

kai yn → x0 (sto A) ¸ste limn f(xn) 6= limn f(yn). An h f tan suneq csto x0, ja èprepe ta dÔo ìria na eÐnai Ðsa me f(x0), �ra kai metaxÔ toucÐsa.


Aplì par�deigma. H sun�rthsh tou Dirichlet, f(x) =

1 x ∈ Q

0 x /∈ Q, eÐnai

asuneq c se k�je x0 ∈ R. Ja d¸soume mia deÔterh apìdeixh, qrhsimopoi¸ntacthn arq thc metafor�c. Apì thn puknìthta twn rht¸n kai twn arr twn, mpo-roÔme na broÔme akoloujÐa (qn) rht¸n arijm¸n me qn → x0 kai akoloujÐa (αn)arr twn arijm¸n me αn → x0. 'Omwc, f(qn) = 1 → 1 kai f(αn) = 0 → 0. Apìthn prohgoÔmenh parat rhsh sumperaÐnoume ìti h f den eÐnai suneq c sto x0.

3.2dþ Sunèqeia kai pr�xeic metaxÔ sunart sewn

To je¸rhma pou akoloujeÐ dÐnei th sqèsh thc sunèqeiac me tic sun jeic alge-brikèc pr�xeic an�mesa se sunart seic. H apìdeix tou eÐnai �mesh, an qrhsimo-poi soume thn arq thc metafor�c se sunduasmì me tic antÐstoiqec idiìthtec giata ìria akolouji¸n.

Je¸rhma 3.2.7. 'Estw f, g : A → R kai èstw x0 ∈ A. Upojètoume ìti oi f, g

eÐnai suneqeÐc sto x0. Tìte,

(i) Oi f + g kai f · g eÐnai suneqeÐc sto x0.

(ii) An epiplèon g(x) 6= 0 gia k�je x ∈ A, tìte h fg orÐzetai sto A kai eÐnai

suneq c sto x0.

Apìdeixh. H apìdeixh ìlwn twn isqurism¸n eÐnai apl : gia par�deigma, gia nadeÐxoume ìti h f

g eÐnai suneq c sto x0, sÔmfwna me thn arq thc metafor�c, arkeÐna deÐxoume ìti, gia k�je akoloujÐa (xn) shmeÐwn tou A pou sugklÐnei sto x0, hakoloujÐa

((fg

)(xn)

)sugklÐnei sto

(fg

)(x0). Apì thn upìjesh, oi f kai g eÐnai

suneqeÐc sto x0. Apì thn arq thc metafor�c èqoume èqoume f(xn) → f(x0) kaig(xn) → g(x0). AfoÔ g(xn) 6= 0 gia k�je n ∈ N kai g(x0) 6= 0, èqoume

(f

g

)(xn) =

f(xn)g(xn)

→ f(x0)g(x0)

=(

f

g

)(x0).

H apìdeixh thc sunèqeiac twn f + g kai f · g sto x0 af netai wc 'Askhsh gia tonanagn¸sth. 2

H stajer sun�rthsh f(x) = c (c ∈ R) kai h tautotik sun�rthsh g(x) = x

eÐnai suneqeÐc sto R. 'Epetai ìti oi poluwnumikèc sunart seic eÐnai suneqeÐc stoR kai ìti k�je rht sun�rthsh eÐnai suneq c se ìla ta shmeÐa tou pedÐou orismoÔthc.

Prìtash 3.2.8 (sÔnjesh suneq¸n sunart sewn). 'Estw f : A → Rkai èstw g : B → R dÔo sunart seic me f(A) ⊆ B. An h f eÐnai suneq c sto x0

kai h g eÐnai suneq c sto f(x0), tìte h g ◦ f : A → R eÐnai suneq c sto x0.

Apìdeixh. 'Estw (xn) akoloujÐa shmeÐwn tou A me xn → x0. AfoÔ h f eÐnaisuneq c sto x0, h arq thc metafor�c deÐqnei ìti f(xn) → f(x0). AfoÔ h g eÐnaisuneq c sto f(x0) ∈ B, gia k�je akoloujÐa (yn) shmeÐwn tou B me yn → f(x0)èqoume g(yn) → g(f(xn)).

'Omwc, f(xn) ∈ B kai f(xn) → f(x0). Sunep¸c,

g(f(xn)) → g(f(x0)).

Gia k�je akoloujÐa (xn) shmeÐwn tou A me xn → x0 deÐxame ìti

(g ◦ f)(xn) = g(f(xn)) → g(f(x0)) = (g ◦ f)(x0).

Apì thn arq thc metafor�c, h g ◦ f eÐnai suneq c sto x0. 2

3.2eþ Sunèqeia kai topik sumperifor�

Apì ton orismì thc sunèqeiac eÐnai fanerì ìti h sumperifor� miac sun�rthshc f

{makri�} apì to x0 den ephre�zei th sunèqeia mh thc f sto x0.

Prìtash 3.2.9. 'Estw f : A → R kai èstw x0 ∈ A. Upojètoume ìti up�rqeiρ > 0 ¸ste o periorismìc thc f sto A ∩ (x0 − ρ, x0 + ρ) na eÐnai sun�rthshsuneq c sto x0. Tìte, h f eÐnai suneq c sto x0.

Apìdeixh. Me ton ìro {periorismìc thc f} ennooÔme th sun�rthsh f̃ : A∩ (x0−ρ, x0 + ρ) → R me f̃(x) = f(x).

'Estw ε > 0. AfoÔ h f̃ eÐnai suneq c sto x0, up�rqei δ1 > 0 ¸ste gia k�jex ∈ (A ∩ (x0 − ρ, x0 + ρ)) me |x− x0| < δ1 na isqÔei |f̃(x)− f̃(xo)| < ε.

Jètoume δ = min{ρ, δ1}. Tìte, èqoume δ > 0 kai an x ∈ A ∩ (x0 − δ, x0 + δ)èqoume tautìqrona x ∈ A ∩ (x0 − ρ, x0 + ρ) kai |x− x0| < δ ≤ δ1. 'Ara,

|f(x)− f(x0)| = |f̃(x)− f̃(xo)| < ε.

Dhlad , h f eÐnai suneq c sto x0. 2

H epìmenh Prìtash deÐqnei ìti an mia sun�rthsh f : A → R eÐnai suneq csto x0 ∈ A, tìte eÐnai {topik� fragmènh}, dhlad fragmènh se mia perioq toux0. Parathr ste ìti mia suneq c sun�rthsh f den eÐnai aparaÐthta fragmènh seolìklhro to pedÐo orismoÔ thc. Apl� paradeÐgmata mac dÐnoun oi sunart seicf(x) = x2 (x ∈ R) kai g(x) = 1

x (x ∈ (0, 1)).

Prìtash 3.2.10. 'Estw f : A → R kai èstw x0 ∈ A. Upojètoume ìti h f

eÐnai suneq c sto x0. Tìte, mporoÔme na broÔme δ > 0 kai M > 0 ¸ste gia k�jex ∈ A ∩ (x0 − δ, x0 + δ) na isqÔei |f(x)| ≤ M .

Apìdeixh. Efarmìzoume ton orismì thc sunèqeiac thc f sto x0 me ε = 1 > 0.Up�rqei δ > 0 ¸ste: an x ∈ A kai |x−x0| < δ, tìte |f(x)−f(x0)| < 1. Dhlad ,gia k�je x ∈ A ∩ (x0 − δ, x0 + δ) èqoume

|f(x)| ≤ |f(x)− f(x0)|+ |f(x0)| < 1 + |f(x0)|.

'Epetai to zhtoÔmeno, me M = 1 + |f(x0)|. 2

H teleutaÐa parat rhsh eÐnai ìti an mia sun�rthsh f : A → R eÐnai suneq csto x0 ∈ A kai an f(x0) 6= 0, tìte h f diathreÐ to prìshmo tou f(x0) se miaolìklhrh (endeqomènwc mikr ) perioq tou x0.

Prìtash 3.2.11. 'Estw f : A → R kai èstw x0 ∈ A. Upojètoume ìti h f

eÐnai suneq c sto x0 kai ìti f(x0) 6= 0.

(i) An f(x0) > 0, tìte up�rqei δ > 0 ¸ste f(x) > 0 gia k�je x ∈ A ∩ (x0 −δ, x0 + δ).

(ii) An f(x0) < 0, tìte up�rqei δ > 0 ¸ste f(x) < 0 gia k�je x ∈ A ∩ (x0 −δ, x0 + δ).

Apìdeixh. Upojètoume pr¸ta ìti f(x0) > 0. AfoÔ h f eÐnai suneq c sto x0, anjewr soume ton ε = f(x0)

2 > 0 up�rqei δ > 0 ¸ste: an x ∈ A kai |x − x0| < δ

tìte

|f(x)− f(x0)| < f(x0)2

=⇒ −f(x0)2

< f(x)− f(x0) =⇒ f(x) > 0.

Dhlad , f(x) > 0 gia k�je x ∈ A ∩ (x0 − δ, x0 + δ).Upojètoume t¸ra ìti f(x0) < 0. AfoÔ h f eÐnai suneq c sto x0, an jew-

r soume ton ε = − f(x0)2 > 0 up�rqei δ > 0 ¸ste: an x ∈ A kai |x − x0| < δ

tìte

|f(x)− f(x0)| < −f(x0)2

=⇒ f(x)− f(x0) < −f(x0)2

=⇒ f(x) < 0.

Dhlad , f(x) < 0 gia k�je x ∈ A ∩ (x0 − δ, x0 + δ). 2

3.3 Basik� jewr mata gia suneqeÐc sunart seic

Se aut n thn par�grafo ja apodeÐxoume dÔo jemeli¸dh kai diaisjhtik� aname-nìmena jewr mata gia suneqeÐc sunart seic pou eÐnai orismènec se èna kleistìdi�sthma: to je¸rhma endi�meshc tim c kai to je¸rhma Ôparxhc mègisthc kaiel�qisthc tim c. H apìdeixh touc apaiteÐ ousiastik qr sh tou axi¸matoc thcplhrìthtac.

3.3aþ To je¸rhma el�qisthc kai mègisthc tim c

To pr¸to basikì je¸rhma mac lèei ìti an f : [a, b] → R eÐnai mia suneq csun�rthsh, tìte h f eÐnai �nw fragmènh kai k�tw fragmènh, kai m�lista paÐrneimègisth kai el�qisth tim . Gia thn apìdeixh ja qrhsimopoi soume thn arq thcmetafor�c.

Je¸rhma 3.3.1. 'Estw f : [a, b] → R suneq c sun�rthsh. Up�rqoun m,M ∈R ¸ste: gia k�je x ∈ [a, b],

m ≤ f(x) ≤ M.

Dhlad , h f eÐnai �nw kai k�tw fragmènh.

Apìdeixh. Ac upojèsoume ìti h f den eÐnai �nw fragmènh. Tìte, gia k�je M > 0mporoÔme na broÔme x ∈ [a, b] me f(x) > M .

3.3 Basika jewrhmata gia suneqeic sunarthseic · 81

Epilègoume diadoqik� M = 1, 2, 3, . . . , n, . . . kai brÐskoume xn ∈ [a, b] me

f(xn) > n, n ∈ N.

JewroÔme thn akoloujÐa (xn). AfoÔ a ≤ xn ≤ b, h (xn) eÐnai fragmènh. Apì toJe¸rhma Bolzano–Weierstrass, up�rqei upakoloujÐa (xkn) thc (xn) me xkn → x0.AfoÔ a ≤ xkn ≤ b gia k�je n, èpetai ìti x0 ∈ [a, b].

H f eÐnai suneq c sto x0 kai xkn→ x0. Apì thn arq thc metafor�c,

f(xkn) → f(x0).

'Ara, h (f(xkn)) eÐnai fragmènh akoloujÐa. 'Omwc, f(xkn) > kn ≥ n apì tontrìpo epilog c twn xkn , dhlad f(xkn) → +∞. Autì eÐnai �topo.

'Omoia deÐqnoume thn Ôparxh k�tw fr�gmatoc ( , pio apl�, jewr ste thn −f :aut eÐnai suneq c, �ra �nw fragmènh sto [a, b] klp). 2

K�nontac èna akìma b ma, deÐqnoume ìti k�je suneq c sun�rthsh f : [a, b] →R paÐrnei mègisth kai el�qisth tim sto [a, b]:

Je¸rhma 3.3.2. 'Estw f : [a, b] → R suneq c sun�rthsh. Up�rqoun y1, y2 ∈[a, b] ¸ste f(y1) ≤ f(x) ≤ f(y2) gia k�je x ∈ [a, b].

Apìdeixh. Apì to Je¸rhma 3.3.1, h f eÐnai �nw fragmènh. Sunep¸c, to sÔnolo

A = {f(x) : x ∈ [a, b]}

eÐnai �nw fragmèno. 'Estw ρ = sup A. Jèloume na deÐxoume ìti up�rqei y2 ∈ [a, b]me f(y2) = ρ.

Apì ton orismì tou supremum, gia k�je n ∈ N mporoÔme na broÔme stoiqeÐotou A sto (ρ− 1/n, ρ]. Dhlad , up�rqei xn ∈ [a, b] gia to opoÐo

ρ− 1n

< f(xn) ≤ ρ.

Apì to krit rio parembol c,f(xn) → ρ.

Apì to Je¸rhma Bolzano–Weierstrass, h akoloujÐa (xn) èqei sugklÐnousa upa-koloujÐa (xkn). Gr�foume y2 gia to limxkn . AfoÔ a ≤ xkn ≤ b gia k�je n,èpetai ìti y2 ∈ [a, b]. AfoÔ h f eÐnai suneq c sto y2, sumperaÐnoume ìti

f(xkn) → f(y2).

Apì th monadikìthta tou orÐou thc (f(xkn)) èpetai ìti

f(y2) = ρ = sup A.

Autì shmaÐnei ìti f(x) ≤ f(y2) gia k�je x ∈ [a, b] (apì ton orismì tou A kai touρ).

Gia thn el�qisth tim ergazìmaste an�loga. 2

DeÔteroc trìpoc. Upojètoume ìti h f den paÐrnei mègisth tim sto sto [a, b].Tìte, f(x) < ρ gia k�je x ∈ [a, b], ìpou ρ = sup{f(x) : x ∈ [a, b]} ìpwc sthnprohgoÔmenh apìdeixh. Sunep¸c, mporoÔme na orÐsoume g : [a, b] → R me

g(x) =1

ρ− f(x).

H g eÐnai suneq c sto [a, b], all� den eÐnai fragmènh. Pr�gmati, sthn prohgoÔ-menh apìdeixh, qrhsimopoi same to gegonìc ìti up�rqei akoloujÐa (xn) sto [a, b]¸ste ρ− 1

n < f(xn). Tìte,

g(xn) =1

ρ− f(xn)> n.

'Eqoume loipìn katal xei se �topo: apì to Je¸rhma 3.3.1, h g èprepe na eÐnai�nw fragmènh. 2

3.3bþ To je¸rhma endi�meshc tim c

Ac upojèsoume ìti mia suneq c sun�rthsh f : [a, b] → R paÐrnei eterìshmectimèc sta �kra tou [a, b]. Tìte, autì pou perimènei kaneÐc apì thn grafik par�-stash thc f eÐnai ìti gia k�poio shmeÐo ξ ∈ (a, b) ja isqÔei f(ξ) = 0 (h kampÔlhy = f(x) ja tm sei ton orizìntio �xona). Ja d¸soume treic apodeÐxeic: ìlec qrh-simopoioÔn ousiastik� to axÐwma thc plhrìthtac. KajemÐa apì autèc {stoqeÔei}se {diaforetik rÐza thc exÐswshc f(x) = 0}.

Je¸rhma 3.3.3. 'Estw f : [a, b] → R suneq c sun�rthsh. Upojètoume ìtif(a) < 0 kai f(b) > 0. Tìte, up�rqei ξ ∈ (a, b) ¸ste f(ξ) = 0.

Pr¸th apìdeixh. Ja prospaj soume na {broÔme} th mikrìterh lÔsh thcexÐswshc f(x) = 0 sto (a, b). Y�qnoume dhlad gia k�poio ξ ∈ (a, b) gia toopoÐo f(ξ) = 0 kai f(x) < 0 gia k�je x me a ≤ x < ξ.

H idèa eÐnai ìti autì to ξ prèpei na eÐnai to supremum tou sunìlou ìlwn twny ∈ (a, b) pou ikanopoioÔn to ex c:

gia k�je x me a ≤ x < y isqÔei f(x) < 0.

OrÐzoume loipìn

A = {y ∈ (a, b] : a ≤ x < y =⇒ f(x) < 0}.

Isqurismìc 1. To A eÐnai mh kenì kai �nw fragmèno.Apìdeixh. EÐnai safèc ìti o b eÐnai �nw fr�gma gia to A. Gia na deÐxoume ìti toA eÐnai mh kenì, skeftìmaste wc ex c: h f eÐnai suneq c sto a kai f(a) < 0.Apì thn Prìtash 3.2.11, up�rqei 0 < δ < b − a ¸ste h f na paÐrnei arnhtikèctimèc sto [a, b] ∩ (a− δ, a + δ) = [a, a + δ). An loipìn a < y < a + δ, tìte

gia k�je x me a ≤ x < y isqÔei f(x) < 0,

to opoÐo shmaÐnei ìti y ∈ A. 'Ara, (a, a + δ) ⊆ A (to A eÐnai mh kenì). 2

Apì to axÐwma thc plhrìthtac up�rqei o ξ = sup A. EpÐshc, a < ξ diìti (a, a +δ) ⊆ A.

Isqurismìc 2. Gia ton ξ = sup A isqÔoun oi a < ξ 0 kai h f eÐnai suneq csto b. Qrhsimopoi¸ntac thn Prìtash 3.2.11, brÐskoume 0 < δ1 < b− a ¸ste giak�je x ∈ (b− δ1, b] na èqoume f(x) > 0. Tìte, o b− δ1 eÐnai �nw fr�gma tou A.Pr�gmati, an y ∈ A tìte f(x) < 0 gia k�je x ∈ [a, y) kai afoÔ f(x) > 0 sto(b− δ1, b] èqoume y ≤ b− δ1. Sunep¸c,

a < a + δ ≤ ξ ≤ b− δ1 < b.

Eidikìtera, a < ξ < b.Mènei na deÐxoume ìti f(ξ) = 0. Ja apokleÐsoume ta endeqìmena f(ξ) < 0

kai f(ξ) > 0.

(i) 'Estw ìti f(ξ) < 0. Apì th sunèqeia thc f sto ξ, up�rqei 0 < δ2 <

min{ξ − a, b − ξ} ¸ste f(x) < 0 sto (ξ − δ2, ξ + δ2) (exhg ste giatÐ).'Omwc tìte, f(x) < 0 sto [a, ξ + δ2) (giatÐ up�rqei y ∈ A me y > ξ − δ2,opìte f(x) < 0 sto [a, y) ∪ (ξ − δ2, ξ + δ2) = [a, ξ + δ2)). Epomènwc,ξ + δ2 ∈ A. Autì eÐnai �topo afoÔ ξ = sup A.

(ii) 'Estw ìti f(ξ) > 0. Tìte, up�rqei 0 < δ3 < min{ξ − a, b − ξ} ¸stef(x) > 0 sto (ξ − δ3, ξ + δ3). An p�roume y ∈ A me y > ξ − δ3 kai z mey > z > ξ − δ3, tìte

y ∈ A =⇒ f(z) < 0

en¸z ∈ (ξ − δ3, ξ + δ3) =⇒ f(z) > 0

dhlad odhgoÔmaste se �topo. 2

Me thn apìdeixh tou deÔterou isqurismoÔ oloklhr¸netai kai h apìdeixh touJewr matoc. 2

DeÔterh apìdeixh. Ja prospaj soume na {broÔme} th megalÔterh lÔshthc exÐswshc f(x) = 0 sto (a, b). Y�qnoume dhlad gia k�poio ξ ∈ (a, b) gia toopoÐo f(ξ) = 0 kai f(x) > 0 gia k�je x me ξ < x ≤ b.

H idèa eÐnai ìti autì to ξ prèpei na eÐnai to supremum tou sunìlou

A = {y ∈ [a, b] : f(y) ≤ 0}.

Isqurismìc 1. To A eÐnai mh kenì kai �nw fragmèno.

Apìdeixh. EÐnai safèc ìti o b eÐnai �nw fr�gma gia to A. To A eÐnai mh kenì:afoÔ f(a) < 0, èqoume a ∈ A. 2

Apì to axÐwma thc plhrìthtac up�rqei o ξ = sup A. EpÐshc, a < ξ. Pr�gmati,sthn prohgoÔmenh apìdeixh eÐdame ìti up�rqei 0 < δ < b−a ¸ste (a, a+δ) ⊆ A.

Isqurismìc 2. Gia ton ξ = sup A isqÔoun oi a < ξ < b kai f(ξ) = 0.

Apìdeixh. DeÐqnoume pr¸ta ìti ξ < b: 'Eqoume f(b) > 0 kai h f eÐnai suneq csto b. Qrhsimopoi¸ntac thn Prìtash 3.2.11, brÐskoume 0 < δ1 < b− a ¸ste giak�je x ∈ (b− δ1, b] na èqoume f(x) > 0. Tìte, o b− δ1 eÐnai �nw fr�gma tou A.Pr�gmati, an y ∈ A tìte f(y) ≤ 0, �ra y ∈ [a, b− δ1]. 'Epetai ìti

ξ = sup A ≤ b− δ1 < b.

Mènei na deÐxoume ìti f(ξ) = 0. Ja deÐxoume ìti f(ξ) ≤ 0 kai f(ξ) ≥ 0 qrhsimo-poi¸ntac thn arq thc metafor�c.

(i) AfoÔ ξ = sup A, up�rqei akoloujÐa (xn) shmeÐwn tou A me xn → ξ.'Eqoume f(xn) ≤ 0 kai h f eÐnai suneq c sto ξ. Apì thn arq thc metafor�cpaÐrnoume f(ξ) = limn f(xn) ≤ 0.

(ii) AfoÔ ξ < b, up�rqei gnhsÐwc fjÐnousa akoloujÐa (yn) sto (ξ, b] me yn → ξ.Gia k�je n ∈ N èqoume yn /∈ A, kai sunep¸c, f(yn) > 0. Apì thn arq thcmetafor�c paÐrnoume f(ξ) = limn f(yn) ≥ 0. 2

Me thn apìdeixh tou deÔterou isqurismoÔ oloklhr¸netai kai h apìdeixh touJewr matoc. 2

TrÐth apìdeixh. ProspajoÔme na proseggÐsoume mia rÐza x0 thc f {opoud -pote} an�mesa sta a kai b, me diadoqikèc diqotom seic tou [a, b]. H Ôparxh thcrÐzac ja exasfalisteÐ apì thn arq twn kibwtismènwn diasthm�twn kai thn arq thc metafor�c.

Sto pr¸to b ma, diqotomoÔme to [a, b] jewr¸ntac to mèso tou a+b2 . An sumbeÐ

na èqoume f(

a+b2

)= 0, jètoume ξ = a+b

2 kai èqoume f(ξ) = 0. Alli¸c, up�rqoundÔo endeqìmena. EÐte f(a+b

2 ) > 0 opìte jètoume a1 = a kai b1 = a+b2 , ,

f(a+b2 ) < 0, opìte jètoume a1 = a+b

2 kai b1 = b. Se k�je perÐptwsh, èqoumef(a1) < 0 kai f(b1) > 0. Parathr ste epÐshc ìti a ≤ a1 < b1 ≤ b kai ìti tom koc tou [a1, b1] eÐnai Ðso me b−a

2 .Epanalamb�noume th diadikasÐa sto [a1, b1]. An f

(a1+b1

2

)= 0, jètoume ξ =

a1+b12 kai èqoume f(ξ) = 0. Alli¸c, brÐskoume a2, b2 pou ikanopoioÔn tic a1 ≤

a2 < b2 ≤ b1, f(a1) < 0, f(b1) > 0 kai b2 − a2 = b−a22 .

SuneqÐzontac epagwgik�, eÐte brÐskoume ξ ∈ [a, b] me f(ξ) = 0 orÐzoumeakoloujÐec (an) kai (bn) sto [a, b] me tic ex c idiìthtec:

(i) a ≤ a1 ≤ a2 ≤ · · · ≤ an ≤ an+1 < bn+1 ≤ bn ≤ · · · ≤ b2 ≤ b1 ≤ b gia k�jen ∈ N.

(ii) f(an) < 0 < f(bn) gia k�je n ∈ N.

(iii) bn − an = b−a2n gia k�je n ∈ N.

H akoloujÐa (an) pou kataskeu�same eÐnai aÔxousa kai �nw fragmènh kai h(bn) eÐnai fjÐnousa kai k�tw fragmènh. 'Ara, sugklÐnoun. Apì thn bn − an =b−a2n → 0, èpetai ìti

limn→∞

an = limn→∞

bn = ξ

gia k�poio ξ ∈ [a, b]. AfoÔ f(an) < 0 kai f(bn) > 0, apì th sunèqeia thc f kaiapì thn arq thc metafor�c paÐrnoume

f(ξ) = limn→∞

f(an) ≤ 0 ≤ limn→∞

f(bn) = f(ξ),

dhlad , f(ξ) = 0. 2

San pìrisma paÐrnoume to je¸rhma endi�meshc tim c:

Je¸rhma 3.3.4. 'Estw f : [a, b] → R suneq c sun�rthsh. An f(a) < f(b) kaif(a) < ρ < f(b)), tìte up�rqei ξ ∈ (a, b) ¸ste f(ξ) = ρ. 'Omoia, an f(b) < f(a)kai f(b) < ρ < f(a)), tìte up�rqei ξ ∈ (a, b) ¸sste f(ξ) = ρ.

Apìdeixh. JewroÔme thn g(x) = f(x) − ρ. H g eÐnai suneq c sto [a, b] kaig(a) = f(a) − ρ < 0, g(b) = f(b) − ρ > 0. Apì to Je¸rhma 3.3.3 up�rqeiξ ∈ (a, b) me g(ξ) = 0, dhlad f(ξ) = ρ.

Gia thn �llh perÐptwsh, qrhsimopoi ste th suneq sun�rthsh h(x) = ρ −f(x). 2

Orismìc 3.3.5 (di�sthma). 'Ena uposÔnolo I tou R lègetai di�sthma angia k�je x, y ∈ I me x < y olìklhro to eujÔgrammo tm ma [x, y] perièqetai stoI.

Me �lla lìgia, diast mata eÐnai ta anoikt�, kleist� hmianoikt� diast matakai oi anoiktèc kleistèc hmieujeÐec.

Je¸rhma 3.3.6. 'Estw I èna di�sthma sto R kai èstw f : I → R suneq csun�rthsh. Tìte, h eikìna f(I) thc f eÐnai di�sthma.

Apìdeixh. 'Estw u, v ∈ f(I) me u < v kai èstw u < w < v. Up�rqoun x, y ∈ I

¸ste f(x) = u kai f(y) = v. QwrÐc periorismì thc genikìthtac mporoÔme naupojèsoume ìti x < y. AfoÔ to I eÐnai di�sthma, èqoume [x, y] ⊆ I kai hf : [x, y] → R eÐnai suneq c sto [x, y]. AfoÔ f(x) = u < w < v = f(y), up�rqeiz ∈ (x, y) ¸ste f(z) = w. AfoÔ z ∈ I, sumperaÐnoume ìti w = f(z) ∈ f(I). Apìton orismì tou diast matoc èpetai ìti to f(I) eÐnai di�sthma. 2

Pìrisma 3.3.7. 'Estw f : [a, b] → R suneq c sun�rthsh. Up�rqoun m ≤ M

sto R ¸ste f([a, b]) = [m,M ].

Apìdeixh. H f eÐnai suneq c kai orÐzetai sto kleistì di�sthma [a, b]. Sunep¸c,h f paÐrnei el�qisth tim m kai mègisth tim M sto [a, b]. Dhlad , m,M ∈f([a, b]) kai f([a, b]) ⊆ [m,M ]. Apì to prohgoÔmeno je¸rhma, to f([a, b]) eÐnaidi�sthma kai perièqei ta m,M . 'Ara, f([a, b]) ⊇ [m,M ]. 'Epetai ìti f([a, b]) =[m, M ]. 2

3.3gþ ParadeÐgmata

H sunèqeia thc f all� kai h upìjesh ìti to pedÐo orismoÔ eÐnai kleistì di�sthmaeÐnai aparaÐthtec sta prohgoÔmena jewr mata.

(a) JewroÔme th sun�rthsh f(x) =

1− |x| x 6= 0

0 x = 0sto [−1, 1]. H f den

paÐrnei mègisth tim sto [−1, 1]. 'Eqoume 1 = sup{f(x) : x ∈ [−1, 1]}, all� o 1den eÐnai tim thc f : parathr ste ìti 0 ≤ f(x) < 1 gia k�je x ∈ [−1, 1]. H f

eÐnai asuneq c sto shmeÐo 0.

(b) JewroÔme thn sun�rthsh f(x) =

−1 0 ≤ x ≤ 1

1 1 < x ≤ 2. Tìte, f(0) < 0 kai

f(2) > 0, all� den up�rqei lÔsh thc f(x) = 0 sto [0, 2]. H f eÐnai asuneq c stoshmeÐo 1.(g) JewroÔme thn f(x) = 1/x sto (0, 1]. H f eÐnai suneq c sto (0, 1], all� deneÐnai �nw fragmènh. To pedÐo orismoÔ thc f den eÐnai kleistì di�sthma.(d) JewroÔme thn f(x) = x sto (0, 1). H f eÐnai suneq c kai fragmènh sto(0, 1), all� den paÐrnei mègisth oÔte el�qisth tim . To pedÐo orismoÔ thc f deneÐnai kleistì di�sthma.

3.3dþ Efarmogèc twn basik¸n jewrhm�twn

To je¸rhma endi�meshc tim c qrhsimopoieÐtai suqn� gia thn apìdeixh thc ÔparxhcrÐzac k�poiac exÐswshc. To pr¸to mac par�deigma eÐnai h {Ôparxh n�ost c rÐzac}pou eÐqame exasfalÐsei me qr sh tou axi¸matoc thc plhrìthtac.

Je¸rhma 3.3.8. 'Estw n ≥ 2 kai èstw ρ ènac jetikìc pragmatikìc arijmìc.Up�rqei monadikìc ξ > 0 ¸ste ξn = ρ.

Apìdeixh. 'Estw ρ > 0. JewroÔme th suneq sun�rthsh f : [0, +∞) → R mef(x) = xn. Pr¸ta ja deÐxoume ìti up�rqei b > 0 ¸ste f(b) > ρ. DiakrÐnoumetreÐc peript¸seic:

(i) An ρ < 1, tìte f(1) = 1n = 1 > ρ.

(ii) An ρ > 1, tìte f(ρ) = ρn > ρ.

(iii) An ρ = 1, tìte f(2) = 2n > 2 > 1.

DeÐxame ìti up�rqei b > 0 ¸ste f(0) = 0 < ρ < f(b). Apì to je¸rhma endi�meshctim c, up�rqei ξ ∈ (0, b) ¸ste f(ξ) = ρ, dhlad ξn = ρ.

H monadikìthta eÐnai apl : èqoume deÐ ìti an a, b > 0 tìte an = bn an kaimìno an a = b. An loipìn èqoume ξn

1 = ρ = ξn2 gia k�poiouc ξ1, ξ2 > 0, tìte

ξ1 = ξ2. 2

Je¸rhma 3.3.9. K�je polu¸numo perittoÔ bajmoÔ èqei toul�qiston mÐa prag-matik rÐza.

Apìdeixh. 'Estw P (x) = amxm + am−1xm−1 + . . . + a1x + a0, ìpou am 6= 0 kai

m perittìc. Gr�foume P (x) = amxm + · · ·+ a1x + a0 = amxm(1 + ∆(x)) ìpou

∆(x) =am−1x

m−1 + · · ·+ a1x + a0

amxm.

Parathr ste ìti an

|x| > 2|am−1|+ · · ·+ |a1|+ |a0|

|am| + 1

tìte |x|k ≤ |x|m−1 gia k�je k = 0, 1, . . . , m− 1, kai sunep¸c,

|∆(x)| ≤ |am−1|xm−1 + · · ·+ |a1|x + |a0||am| |x|m

≤ |am−1| |x|m−1 + · · ·+ |a1| |x|m−1 + |a0| |x|m−1

|am| |x|m

=|am−1|+ · · ·+ |a1|+ |a0|

|am| |x|<

12.

'Ara, up�rqei M > 0 ¸ste an |x| ≥ M tìte

1 + ∆(x) ≥ 1− |∆(x)| ≥ 1− 12

=12

> 0.

Dhlad , an |x| ≥ M tìte oi P (x) kai amxm èqoun to Ðdio prìshmo. 'Epetaiìti o P (−M)P (M) eÐnai omìshmoc me ton a2

m(−M)mMm = a2mM2m(−1)m,

dhlad arnhtikìc. Apì to je¸rhma endi�meshc tim c up�rqei ξ ∈ (−M, M) ¸steP (ξ) = 0. 2

Je¸rhma 3.3.10 (je¸rhma stajeroÔ shmeÐou). 'Estw f : [0, 1] →[0, 1] suneq c sun�rthsh. Up�rqei x0 ∈ [0, 1] ¸ste f(x0) = x0.

Apìdeixh. Jèloume na deÐxoume ìti h kampÔlh y = f(x) tèmnei thn diag¸nioy = x. ArkeÐ na deÐxoume ìti h suneq c sun�rthsh h(x) = f(x)− x mhdenÐzetaik�pou sto [0, 1].

An f(0) = 0 f(1) = 1 èqoume to zhtoÔmeno gia x0 = 0 x0 = 1 antÐstoiqa.Upojètoume loipìn ìti f(0) > 0 kai f(1) < 1. Tìte, h(0) = f(0) > 0 kai

h(1) = f(1)−1 < 0. Apì to je¸rhma endi�meshc tim c, up�rqei x0 ∈ (0, 1) ¸steh(x0) = 0. Dhlad , f(x0) = x0. 2

3.3eþ Sunèqeia antÐstrofhc sun�rthshc

'Estw I èna di�sthma sto R. Xekin�me apì thn parat rhsh ìti mia 1-1 sun�rthshf : I → R den eÐnai upoqrewtik� monìtonh. P�rte gia par�deigma thn f : (0, 2) →

R pou orÐzetai apì thn f(x) =

4− x 0 < x < 12 x = 1

x− 1 1 < x < 2. H f eÐnai 1-1, ìmwc

eÐnai fjÐnousa sto (0, 1) kai aÔxousa sto (1, 2).To je¸rhma pou akoloujeÐ deÐqnei ìti an mia suneq c sun�rthsh f : I → R

eÐnai èna proc èna, tìte eÐnai gnhsÐwc monìtonh.

Je¸rhma 3.3.11. 'Estw f : I → R suneq c kai 1-1 sun�rthsh. Tìte, h f

eÐnai gnhsÐwc aÔxousa gnhsÐwc fjÐnousa sto I.

Pr¸th apìdeixh. Ja k�noume thn apìdeixh se trÐa b mata.

B ma 1. An a, b, c ∈ I me a f(b) >

f(c).

Apìdeixh. AfoÔ h f eÐnai 1-1, oi f(a), f(b) kai f(c) eÐnai diaforetikoÐ an� dÔo.MporoÔme na upojèsoume ìti f(a) < f(b) (alli¸c, jewroÔme thn −f). JadeÐxoume ìti f(a) < f(b) < f(c), apokleÐontac tic peript¸seic f(c) < f(a) kaif(a) < f(c) < f(b).

(i) An f(c) < f(a) < f(b) tìte apì to je¸rhma endi�meshc tim c up�rqeix ∈ (b, c) me f(x) = f(a), to opoÐo eÐnai �topo, afoÔ a < x kai h f eÐnai1-1.

(ii) An f(a) < f(c) < f(b) tìte apì to je¸rhma endi�meshc tim c up�rqeiy ∈ (a, b) me f(y) = f(c), to opoÐo eÐnai epÐshc �topo, afoÔ y < c kai h f

eÐnai 1-1.

B ma 2. An a, b, c, d ∈ I me a f(b) > f(c) > f(d).

Apìdeixh. MporoÔme na upojèsoume ìti f(a) < f(b). Efarmìzontac to B ma 1gia thn tri�da a, b, c blèpoume ìti f(a) < f(b) < f(c). Efarmìzontac xan� toB ma 1 gia thn tri�da b, c, d blèpoume ìti f(b) < f(c) < f(d). Dhlad ,

f(a) < f(b) < f(c) < f(d).

Xekin¸ntac apì thn upìjesh ìti f(a) > f(b), deÐqnoume me ton Ðdio trìpo ìti

f(a) > f(b) > f(c) > f(d).

B ma 3. StajeropoioÔme dÔo shmeÐa a < b sto I. MporoÔme na upojèsoumeìti f(a) < f(b). Ja deÐxoume ìti h f eÐnai gnhsÐwc aÔxousa, deÐqnontac ìti anx, y ∈ I kai x < y, tìte f(x) < f(y).

An x = a kai y = b, tìte f(x) = f(a) < f(b) = f(y). Alli¸c, an�loga methn di�taxh twn x, y sthn tetr�da a, b, x, y, to B ma 2 ( to B ma 1 an x = a y = b) deÐqnei ìti h Ðdia di�taxh ja isqÔei gia tic eikìnec f(x), f(y) sthn tetr�daf(a), f(b), f(x), f(y). Gia par�deigma, an x < a < b < y tìte f(x) < f(a) <

f(b) < f(y), �ra f(x) < f(y). An a < x = b < y tìte f(a) < f(x) = f(b) <

f(y), �ra f(x) < f(y). 2

DeÔterh apìdeixh. Epilègoume tuqìnta shmeÐa x0 < y0 sto I. 'Eqoumef(x0) 6= f(y0), �ra f(x0) < f(y0) f(x0) > f(y0). Upojètoume ìti f(x0) <

f(y0) kai ja deÐxoume ìti h f eÐnai gnhsÐwc aÔxousa (ìmoia deÐqnoume ìti anf(x0) > f(y0) tìte h f eÐnai gnhsÐwc fjÐnousa).

'Estw x1, y1 ∈ I me x1 < y1. Jèloume na deÐxoume ìti f(x1) < f(y1).JewroÔme tic sunart seic h, g : [0, 1] → I me

h(t) = (1− t)x0 + tx1 kai g(t) = (1− t)y0 + ty1.

Parathr ste ìti h(t), g(t) ∈ I gia k�je t ∈ [0, 1] (kaj¸c to t diatrèqei to [0, 1],to h(t) kineÐtai apì to x0 proc to x1 kai to g(t) kineÐtai apì to y0 proc to y1 �afoÔ to I eÐnai di�sthma, ta h(t), g(t) mènoun mèsa s� autì).

Parathr ste epÐshc ìti, lìgw twn x0 < y0 kai x1 < y1 èqoume

h(t) = (1− t)x0 + tx1 < (1− t)y0 + ty1 = g(t)

gia k�je t ∈ [0, 1]. OrÐzoume H(t) = f(h(t)) − f(g(t)). H H eÐnai suneq csun�rthsh wc diafor� sunjèsewn suneq¸n sunart sewn. AfoÔ h(t) 6= g(t) kaih f eÐnai 1-1, paÐrnoume

H(t) 6= 0

gia k�je t ∈ [0, 1]. 'Omwc, H(0) = f(h(0)) − f(g(0)) = f(x0) − f(y0) < 0 apìthn upìjesh mac. 'Epetai ìti H(t) < 0 gia k�je t ∈ [0, 1]: an h H èpairne k�poujetik tim , tìte ja èpairne kai thn tim 0 apì to je¸rhma endi�meshc tim c(�topo). 'Eqoume loipìn

f(h(t)) < f(g(t)) gia k�je t ∈ [0, 1].

Eidikìtera, f(h(1)) < f(g(1)), dhlad f(x1) < f(y1). 2

Den eÐnai dÔskolo na perigr�yei kaneÐc thn eikìna f(I) miac suneqoÔc kai 1-1sun�rthshc f : I → R. Ac upojèsoume gia par�deigma ìti to I eÐnai èna kleistìdi�sthma [a, b] kai ìti h f eÐnai gnhsÐwc aÔxousa (apì to Je¸rhma 3.3.11 h f eÐnaignhsÐwc monìtonh). Tìte, h eikìna thc f eÐnai to kleistì di�sthma [f(a), f(b)].An to I eÐnai di�sthma anoiktì se k�poio kai sta dÔo apì ta �kra tou ( di�sthma me �kro k�poio apì ta ±∞), tìte, ìpwc eÐdame, h eikìna f(I) thc f

eÐnai k�poio di�sthma. OrÐzoume thn antÐstrofh sun�rthsh f−1 : f(I) → I wcex c: an y ∈ f(I), up�rqei monadikì x ∈ I ¸ste f(x) = y. Jètoume f−1(y) = x.Dhlad ,

f−1(y) = x ⇐⇒ f(x) = y.

Parathr ste ìti h f−1 èqei thn Ðdia monotonÐa me thn f . Gia par�deigma, acupojèsoume ìti h f eÐnai gnhsÐwc aÔxousa. 'Estw y1, y2 ∈ f(I) me y1 < y2. An tan f−1(y1) ≥ f−1(y2), tìte ja eÐqame

f(f−1(y1)) ≥ f(f−1(y2)), dhlad y1 ≥ y2.

Autì eÐnai �topo, �ra f−1(y1) < f−1(y2).Ja deÐxoume ìti h antÐstrofh suneqoÔc kai 1-1 sun�rthshc eÐnai epÐshc su-

neq c.

Je¸rhma 3.3.12. 'Estw f : I → R suneq c kai 1-1 sun�rthsh. Tìte, hf−1 : f(I) → R eÐnai suneq c.

Apìdeixh. MporoÔme na upojèsoume ìti h f eÐnai gnhsÐwc aÔxousa. 'Estwy0 ∈ f(I). Upojètoume ìti to y0 den eÐnai �kro tou f(I) (oi �llec peript¸seicelègqontai ìmoia). Tìte, y0 = f(x0) gia k�poio eswterikì shmeÐo tou I.

'Estw ε > 0. MporoÔme na upojèsoume ìti x0 − ε, x0 + ε ∈ I (oÔtwc �llwc, gia na elègxoume th sunèqeia mac endiafèroun ta mikr� ε > 0). Jèloumena broÔme δ > 0 ¸ste

|y − y0| < δ kai y ∈ f(I) =⇒ |f−1(y)− x0| < ε.

Gia thn epilog tou δ douleÔoume wc ex c: afoÔ f(x0 − ε) < y = f(x0) <

f(x0 + ε), up�rqoun δ1, δ2 > 0 ¸ste f(x0− ε) = y0− δ1 kai f(x0 + ε) = y0 + δ2.Epilègoume δ = min{δ1, δ2} > 0.

An |y− y0| < δ, tìte f(x0− ε) < y < f(x0 + ε). Apì to je¸rhma endi�meshctim c, up�rqei x ∈ I ¸ste f(x) = y. To x eÐnai monadikì giatÐ h f eÐnai 1-1, kai

x0 − ε < x < x0 + ε

giatÐ h f−1 eÐnai gnhsÐwc aÔxousa. 'Ara, |f−1(y) − f−1(y0)| = |x − x0| < ε.Dhlad , h f−1 eÐnai suneq c sto y0. 2

Kef�laio 4

'Oria sunart sewn

4.1 ShmeÐa suss¸reushc

Orismìc 4.1.1. 'Estw A èna mh kenì uposÔnolo tou R kai èstw x0 ∈ R.Lème ìti o x0 eÐnai shmeÐo suss¸reushc tou A an gia k�je δ > 0 mporoÔmena broÔme x ∈ A ¸ste 0 < |x − x0| < δ (isodÔnama: x ∈ (x0 − δ, x0 + δ) kaix 6= x0).

Dhlad , o x0 eÐnai shmeÐo suss¸reushc tou A an osod pote kont� ston x0

mporoÔme na broÔme stoiqeÐa tou A diaforetik� apì ton x0. Parathr ste ìtiden apaitoÔme apì ton x0 na eÐnai stoiqeÐo tou A.

ParadeÐgmata 4.1.2. (a) An A = [a, b], tìte o x0 eÐnai shmeÐo suss¸reushctou A an kai mìno an x0 ∈ [a, b]. An A = (a, b], tìte k�je shmeÐo tou A eÐnaishmeÐo suss¸reushc tou A, kai up�rqei �llo èna shmeÐo suss¸reushc tou A, toa, to opoÐo den an kei sto sÔnolo.(b) An A = [0, 1] ∪ {2}, tìte 2 ∈ A all� o 2 den eÐnai shmeÐo suss¸reushc touA.(g) To N = {1, 2, 3, . . .} den èqei kanèna shmeÐo suss¸reushc.(d) An A = {1, 1

2 , 13 , . . .}, tìte o 0 eÐnai to monadikì shmeÐo suss¸reushc tou A

(kai den an kei sto A).

Orismìc 4.1.3. 'Estw A èna mh kenì uposÔnolo tou R kai èstw x0 ∈ A. Lèmeìti o x0 eÐnai memonwmèno shmeÐo tou A an den eÐnai shmeÐo suss¸reushctou A, dhlad , an up�rqei perioq tou x0 h opoÐa den perièqei �lla shmeÐa tou A

ektìc apì to x0 (isodÔnama, an up�rqei δ > 0 ¸ste A∩ (x0− δ, x0 + δ) = {x0}).H epìmenh Prìtash dÐnei qr simouc qarakthrismoÔc tou shmeÐou suss¸reu-

shc.

Prìtash 4.1.4. 'Estw A èna mh kenì uposÔnolo tou R kai èstw x0 ∈ R. Taex c eÐnai isodÔnama:

(i) To x0 eÐnai shmeÐo suss¸reushc tou A.

(ii) Gia k�je δ > 0 up�rqoun �peira to pl joc shmeÐa tou A sto (x0−δ, x0+δ).

92 · Oria sunarthsewn

(iii) Up�rqei akoloujÐa (xn) diaforetik¸n an� dÔo, kai diaforetik¸n apì to x0,shmeÐwn tou A, h opoÐa sugklÐnei sto x0.

Apìdeixh. (i)⇒(ii) 'Estw δ > 0. AfoÔ to x0 eÐnai shmeÐo suss¸reushc tou A,sto (x0 − δ, x0 + δ) mporoÔme na broÔme shmeÐa tou A diaforetik� apì to x0.Ac upojèsoume ìti aut� ta shmeÐa eÐnai peperasmèna to pl joc, ta y1, . . . , ym.K�poio apì aut�, ac poÔme to yj gia k�poion 1 ≤ j ≤ m, eÐnai to plhsièsteroproc to x0. Jètoume δ1 = |x0−yj |. Tìte, δ1 > 0 (diìti yj 6= x0) kai sthn perioq (x0−δ1, x0+δ1) den up�rqei shmeÐo tou A diaforetikì apì to x0 (exhg ste giatÐ).Autì eÐnai �topo, diìti to x0 eÐnai shmeÐo suss¸reushc tou A.

(ii)⇒(iii) Apì thn upìjesh, sto (x0 − 1, x0 + 1) up�rqoun �peira to pl jocshmeÐa tou A. MporoÔme loipìn na broÔme x1 ∈ A me x1 6= x0 kai |x1 − x0| < 1.

OmoÐwc, sto(x0 − 1

2 , x0 + 12

)up�rqoun �peira to pl joc shmeÐa tou A. Mpo-

roÔme loipìn na broÔme x2 ∈ A me x2 6= x0, x1 kai |x2 − x0| < 12 .

SuneqÐzoume me ton Ðdio trìpo: èstw ìti èqoume breÐ x1, x2, . . . , xn−1 ∈ A

diaforetik� an� dÔo (kai diaforetik� apì to x0) ¸ste |xk − x0| < 1k gia k�je

k = 1, 2, . . . , n− 1. Sto(x0 − 1

n , x0 + 1n

)up�rqoun �peira to pl joc shmeÐa tou

A. MporoÔme loipìn na broÔme xn ∈ A me xn 6= x0, x1, . . . , xn−1 kai |xn−x0| <1n .

H akoloujÐa (xn), pou orÐzetai me autìn ton trìpo, sugklÐnei sto x0 kai èqeiìrouc pou an koun sto A, eÐnai diaforetikoÐ an� dÔo kai diaforetikoÐ apì tonx0.

(iii)⇒(i) Upojètoume ìti up�rqei akoloujÐa (xn) diaforetik¸n an� dÔo shmeÐwntou A, h opoÐa sugklÐnei sto x0. 'Estw δ > 0. Up�rqei n0 ∈ N ¸ste |xn−x0| < δ

gia k�je n ≥ n0. AfoÔ oi ìroi thc (xn) eÐnai diaforetikoÐ an� dÔo, k�poioc apìautoÔc (gia thn akrÐbeia, �peiroi to pl joc) eÐnai diaforetikìc apì to x0 kaian kei sto (x0 − δ, x0 + δ). AfoÔ to δ > 0 tan tuqìn, to x0 eÐnai shmeÐosuss¸reushc tou A. 2

4.2 Orismìc tou orÐou

Orismìc 4.2.1 (ìrio sun�rthshc). 'Estw f : A → R kai èstw x0 ènashmeÐo suss¸reushc tou A. Lème ìti to ìrio thc f kaj¸c to x teÐnei sto x0

up�rqei kai isoÔtai me ` ∈ R an:

Gia k�je ε > 0 up�rqei δ > 0 ¸ste: an x ∈ A kai 0 < |x − x0| < δ,tìte |f(x)− `| < ε.

An ènac tètoioc arijmìc ` up�rqei, tìte eÐnai monadikìc (deÐxte to) kai gr�foume` = limx→x0 f(x) f(x) → ` kaj¸c x → x0.

Orismìc 4.2.2. 'Estw f : A → R kai èstw x0 èna shmeÐo suss¸reushc touA.(a) Lème ìti h f teÐnei sto +∞ kaj¸c to x → x0 an:

Gia k�je M > 0 up�rqei δ > 0 ¸ste: an x ∈ A kai 0 < |x− x0| < δ

tìte f(x) > M .

4.2 Orismoc tou oriou · 93

Se aut n thn perÐptwsh, gr�foume limx→x0 f(x) = +∞.(b) Lème ìti h f teÐnei sto −∞ kaj¸c to x → x0 an:

Gia k�je M > 0 up�rqei δ > 0 ¸ste: an x ∈ A kai 0 < |x− x0| < δ

tìte f(x) < −M .

Se aut n thn perÐptwsh, gr�foume limx→x0 f(x) = −∞.

Parat rhsh 4.2.3. Parathr ste ìti mporoÔme na exet�soume thn Ôparxh mh tou orÐou thc f : A → R se k�je shmeÐo suss¸reushc x0 tou A. To x0 mporeÐna an kei na mhn an kei sto A: arkeÐ na up�rqoun x ∈ A osod pote kont� stox0. EpÐshc, akìma ki an to x0 an kei sto pedÐo orismoÔ thc f , h tim f(x0) denephre�zei thn Ôparxh mh tou limx→x0 f(x) oÔte kai thn tim tou orÐou (an autìup�rqei).

ParadeÐgmata 4.2.4. (a) limx→3 x2 = 9.

(b) 'Estw f : R→ R me f(x) ={

x2 an x 6= 02 an x = 0

. To limx→0 f(x) up�rqei kai

eÐnai Ðso me 0, en¸ f(0) = 2.(g) 'Estw f : (0, +∞) → R me f(x) = 1

x . Tìte, limx→0 f(x) = +∞. Anjewr soume thn g : R \ {0} → R me f(x) = 1

x , tìte to ìrio limx→0 g(x) denup�rqei.

MporeÐte na apodeÐxete ìlouc autoÔc touc isqurismoÔc me b�sh ton orismì('Askhsh). MporeÐte epÐshc na qrhsimopoi sete thn arq thc metafor�c, thnopoÐa ja suzht soume parak�tw, ¸ste na anaqjeÐte sta antÐstoiqa ìria ako-louji¸n.(d) 'Estw f : [0, 1] → R me f(x) = 0 an x �rrhtoc x = 0, kai f(x) = 1

q an x = pq

me p, q ∈ N kai MKD(p, q) = 1. Tìte, gia k�je x0 ∈ [0, 1] to ìrio limx→x0 f(x)up�rqei kai isoÔtai me 0.

Pr�gmati, èstw x0 ∈ [0, 1] kai èstw ε > 0. Jètoume M = M(ε) =[1ε

]kai

A(ε) = {y ∈ [0, 1] : y 6= x0 kai f(y) ≥ ε}. An o y an kei sto A(ε) tìte eÐnairhtìc o opoÐoc gr�fetai sth morf y = p

q ìpou p, q ∈ N, p ≤ q kai f(y) = 1q ≥ ε.

To pl joc aut¸n twn arijm¸n eÐnai to polÔ Ðso me to pl joc twn zeugari¸n(p, q) fusik¸n arijm¸n ìpou q ≤ M kai p ≤ q. Epomènwc, den xepern�ei tonM(M + 1)/2. Dhlad , to A(ε) eÐnai peperasmèno sÔnolo. MporoÔme loipìn nagr�youme A(ε) = {y1, . . . , ym} ìpou m = m(ε) ∈ N.

O arijmìc δ = min{|x0 − y1|, . . . , |x0 − ym|} eÐnai gn sia jetikìc. 'Estwx ∈ [0, 1] me 0 < |x− x0| < δ. Tìte, x /∈ A(ε), �ra 0 ≤ f(x) < ε. AfoÔ to ε > 0 tan tuqìn, sumperaÐnoume ìti limx→x0 f(x) = 0.

Orismìc 4.2.5 (pleurik� ìria). 'Estw f : A → R kai èstw x0 shmeÐosuss¸reushc tou A ¸ste gia k�je δ > 0 na up�rqoun stoiqeÐa tou A sto(x0 − δ, x0) (èna tètoio x0 lègetai shmeÐo suss¸reushc tou A apì arister�).Lème ìti:

(i) limx→x−0f(x) = ` ∈ R (o ` eÐnai to pleurikì ìrio thc f kaj¸c to

x teÐnei sto x0 apì arister�) an: gia k�je ε > 0 up�rqei δ > 0 ¸stean x ∈ A kai x0 − δ < x < x0 tìte |f(x)− `| < ε.

TeleÐwc an�loga, èstw x0 shmeÐo suss¸reushc tou A ¸ste gia k�je δ > 0 naup�rqoun stoiqeÐa tou A sto (x0, x0 + δ) (èna tètoio x0 lègetai shmeÐo suss¸-reushc tou A apì dexi�). Lème ìti:

(ii) limx→x+0

f(x) = ` ∈ R (o ` eÐnai to pleurikì ìrio thc f kaj¸c tox teÐnei sto x0 apì dexi�) an: gia k�je ε > 0 up�rqei δ > 0 ¸ste anx ∈ A kai x0 < x < x0 + δ tìte |f(x)− `| < ε.

Af noume wc �skhsh gia ton anagn¸sth na d¸sei austhroÔc orismoÔc gia taex c: limx→x−0

f(x) = +∞, limx→x−0f(x) = −∞, limx→x+

0f(x) = +∞ kai

limx→x+0

f(x) = −∞.

Apì ton orismì twn pleurik¸n orÐwn èpetai �mesa h akìloujh Prìtash.

Prìtash 4.2.6. 'Estw f : A → R mia sun�rthsh kai èstw x0 ∈ R shmeÐosuss¸reushc tou A apì arister� kai apì dexi�. Tìte to limx→x0 f(x) up�rqeian kai mìnon an ta dÔo pleurik� ìria limx→x−0

f(x) kai limx→x+0

f(x) up�rqounkai eÐnai Ðsa. 2

Orismìc 4.2.7. 'Estw A èna mh kenì uposÔnolo tou R. Lème ìti to +∞ eÐnaishmeÐo suss¸reushc tou A an gia k�je M > 0 mporoÔme na broÔme x ∈ A ¸stex > M . EÔkola elègqoume ìti autì sumbaÐnei an kai mìno an up�rqei akoloujÐa(xn) sto A me xn → +∞.

AntÐstoiqa, lème ìti to −∞ eÐnai shmeÐo suss¸reushc tou A an gia k�jeM > 0 mporoÔme na broÔme x ∈ A ¸ste x < −M . EÔkola elègqoume ìti autìsumbaÐnei an kai mìno an up�rqei akoloujÐa (xn) sto A me xn → −∞.

Orismìc 4.2.8. 'Estw f : A → R kai èstw ìti to +∞ eÐnai shmeÐo suss¸-reushc tou A.(a) Lème ìti to ìrio thc f kaj¸c to x teÐnei sto +∞ up�rqei kai isoÔtai me ` ∈ Ran:

Gia k�je ε > 0 up�rqei M > 0 ¸ste: an x ∈ A kai x > M , tìte|f(x)− `| < ε.

An ènac tètoioc arijmìc ` up�rqei, tìte eÐnai monadikìc (deÐxte to) kai gr�foume` = limx→+∞ f(x).(b) Lème ìti h f teÐnei sto +∞ kaj¸c to x → +∞ an:

Gia k�je M1 > 0 up�rqei M2 > 0 ¸ste: an x ∈ A kai x > M2 tìtef(x) > M1.

Se aut n thn perÐptwsh, gr�foume limx→+∞ f(x) = +∞.(g) Lème ìti h f teÐnei sto −∞ kaj¸c to x → +∞ an:

Gia k�je M1 > 0 up�rqei M2 > 0 ¸ste: an x ∈ A kai x > M2 tìtef(x) < −M1.

Se aut n thn perÐptwsh, gr�foume limx→+∞ f(x) = −∞.TeleÐwc an�loga, an f : A → R kai an to −∞ eÐnai shmeÐo suss¸reushc

tou A, mporoÔme na orÐsoume kajemÐa apì tic prot�seic limx→−∞ f(x) = `,limx→−∞ f(x) = +∞ kai limx→−∞ f(x) = −∞.

4.2 Orismoc tou oriou · 95

4.2aþ Arq thc metafor�c

'Estw f : A → R kai èstw x0 ∈ R èna shmeÐo suss¸reushc tou A. H arq thc metafor�c dÐnei ènan qarakthrismì thc Ôparxhc tou orÐou thc f kaj¸cto x teÐnei sto x0 mèsw akolouji¸n.

Je¸rhma 4.2.9 (arq thc metafor�c gia to ìrio). 'Estw f : A → Rkai èstw x0 èna shmeÐo suss¸reushc tou A. Tìte, limx→x0 f(x) = ` an kai mìnoan: gia k�je akoloujÐa (xn) shmeÐwn tou A me xn 6= x0 kai xn → x0, h akoloujÐa(f(xn)) sugklÐnei sto `.

Apìdeixh. Upojètoume pr¸ta ìti limx→x0 f(x) = `. 'Estw xn ∈ A me xn 6= x0

kai xn → x0. Ja deÐxoume ìti f(xn) → `: 'Estw ε > 0. AfoÔ limx→x0 f(x) = `,up�rqei δ > 0 ¸ste: an x ∈ A kai 0 < |x− x0| < δ, tìte |f(x)− `| < ε.

'Eqoume upojèsei ìti xn 6= x0 kai xn → x0. 'Ara, gi� autì to δ > 0 mporoÔmena broÔme n0 ∈ N ¸ste: an n ≥ n0 tìte 0 < |xn − x0| < δ.

Sundu�zontac ta parap�nw èqoume: an n ≥ n0, tìte 0 < |xn − x0| < δ �ra

|f(xn)− `| < ε.

AfoÔ to ε > 0 tan tuqìn, f(xn) → `.

Gia thn antÐstrofh kateÔjunsh ja doulèyoume me apagwg se �topo. U-pojètoume ìti gia k�je akoloujÐa (xn) shmeÐwn tou A me xn 6= x0 kai xn →x0, h akoloujÐa (f(xn)) sugklÐnei sto `. Upojètoume epÐshc ìti den isqÔei hlimx→x0 f(x) = ` kai ja katal xoume se �topo.

AfoÔ den isqÔei h limx→x0 f(x) = `, up�rqei k�poio ε > 0 me thn ex cidiìthta:

(∗) Gia k�je δ > 0 up�rqei x ∈ A to opoÐo ikanopoieÐ thn 0 <

|x− x0| < δ all� |f(x)− `| ≥ ε.

QrhsimopoioÔme thn (∗) diadoqik� me δ = 1, 12 , . . . , 1

n , . . .. Gia k�je n ∈ Nèqoume 1/n > 0 kai apì thn (∗) brÐskoume xn ∈ A me 0 < |xn − x0| < 1/n kai|f(xn)−`| ≥ ε. 'Eqoume xn 6= x0 kai apì to krit rio parembol c eÐnai fanerì ìtixn → x0. Apì thn upìjesh pou k�name prèpei h akoloujÐa (f(xn)) na sugklÐneisto `. Autì ìmwc eÐnai adÔnato afoÔ |f(xn)− `| ≥ ε gia k�je n ∈ N. 2

Parathr seic 4.2.10. (a) H arq thc metafor�c mporeÐ na qrhsimopoihjeÐme dÔo diaforetikoÔc trìpouc:

(i) gia na deÐxoume ìti limx→x0 f(x) = ` arkeÐ na deÐxoume ìti {xn 6= x0 kaixn → x0 =⇒ f(xn) → `}.

(ii) gia na deÐxoume ìti den isqÔei h limx→x0 f(x) = ` arkeÐ na broÔme miaakoloujÐa xn → x0 (sto A), me xn 6= x0, ¸ste limn f(xn) 6= `. PolÔsuqn�, exasfalÐzoume k�ti isqurìtero, ìti den up�rqei to limx→x0 f(x),brÐskontac dÔo akoloujÐec xn → x0 kai yn → x0 (sto A), me xn, yn 6= x0,¸ste limn f(xn) 6= limn f(yn). An up rqe to limx→x0 f(x), ja èprepe tadÔo ìria na eÐnai metaxÔ touc Ðsa.


(b) MporoÔme na diatup¸soume kai na apodeÐxoume antÐstoiqec morfèc thc ar-q c thc metafor�c gia touc upìloipouc tÔpouc orÐwn pleurik¸n orÐwn pousuzht same.

To je¸rhma pou akoloujeÐ dÐnei th sqèsh tou orÐou me tic sun jeic algebrikècpr�xeic an�mesa se sunart seic. H apìdeix tou eÐnai �mesh, an qrhsimopoi sou-me thn arq thc metafor�c se sunduasmì me tic antÐstoiqec idiìthtec gia ta ìriaakolouji¸n.

Je¸rhma 4.2.11. 'Estw f, g : A → R kai èstw x0 shmeÐo suss¸reushc touA. Upojètoume ìti up�rqoun ta limx→x0 f(x) = ` kai limx→x0 g(x) = m. Tìte,

(i) limx→x0(f(x) + g(x)) = ` + m kai limx→x0(f(x)g(x)) = ` ·m.

(ii) An epiplèon g(x) 6= 0 gia k�je x ∈ A kai limx→x0 g(x) = m 6= 0, tìte h fg

orÐzetai sto A kai limx→x0f(x)g(x) = `

m .

Apìdeixh. H apìdeixh ìlwn twn isqurism¸n, me apl qr sh thc arq c thc meta-for�c, af netai wc 'Askhsh gia ton anagn¸sth. 2

Prìtash 4.2.12. 'Estw A èna mh kenì uposÔnolo tou R, èstw x0 ∈ R shmeÐosuss¸reushc tou A kai A : X → R. Upojètoume ìti to ìrio limx→x0 f(x) up�rqeikai eÐnai Ðso me `. 'Estw g : B → R me f(A) ⊆ B kai ` ∈ B. An h g eÐnai suneq csto `, tìte to ìrio limx→xo(g ◦ f)(x) up�rqei kai isoÔtai me g(`).

Apìdeixh. 'Estw (xn) akoloujÐa shmeÐwn tou A me xn 6= x0 kai xn → x0. AfoÔlimx→x0 f(x) = `, h arq thc metafor�c deÐqnei ìti f(xn) → `. AfoÔ h g eÐnaisuneq c sto ` ∈ B, gia k�je akoloujÐa (yn) shmeÐwn tou B me yn → ` èqoumeg(yn) → g(`).

'Omwc, f(xn) ∈ B kai f(xn) → `. Sunep¸c,

g(f(xn)) → g(`).

Gia k�je akoloujÐa (xn) shmeÐwn tou A me xn 6= x0 kai xn → x0 deÐxame ìti

(g ◦ f)(xn) = g(f(xn)) → g(`).

Apì thn arq thc metafor�c, sumperaÐnoume ìti limx→xo(g ◦ f)(x) = g(`). 2

4.3 Sqèsh orÐou kai sunèqeiac

Se aut n thn par�grafo ja sundèsoume thn ènnoia tou orÐou me thn ènnoia thcsunèqeiac. Parathr ste ìti h sunèqeia elègqetai se k�je shmeÐo tou pedÐouorismoÔ miac sun�rthshc, èqei loipìn nìhma na exet�soume pr¸ta th sunèqeiasta memonwmèna shmeÐa tou pedÐou orismoÔ. 'Opwc deÐqnei h epìmenh Prìtash,k�je sun�rthsh eÐnai suneq c se ìla aut� ta shmeÐa.

Prìtash 4.3.1. 'Estw f : A → R kai èstw x0 èna memonwmèno shmeÐo touA. Tìte, h f eÐnai suneq c sto x0.

4.3 Sqesh oriou kai suneqeiac · 97

Apìdeixh. AfoÔ to x0 eÐnai memonwmèno shmeÐo tou A, up�rqei δ > 0 ¸steA ∩ (x0 − δ, x0 + δ) = {x0}. Dhlad , an x ∈ A kai |x − x0| < δ, tìte x = x0.Ja deÐxoume ìti {autì to δ douleÔei gia ìla ta ε}.

'Estw ε > 0. An x ∈ A kai |x− x0| < δ, tìte x = x0, kai sunep¸c,

|f(x)− f(x0)| = |f(x0)− f(x0)| = 0 < ε.

Br kame δ > 0 ¸ste gia k�je x ∈ A me |x−x0| < δ na isqÔei |f(x)−f(x0)| < ε.Me b�sh ton orismì thc sunèqeiac, h f eÐnai suneq c sto x0. 2

An to x0 ∈ A eÐnai kai shmeÐo suss¸reushc tou A, tìte h sqèsh orÐou kaisunèqeiac dÐnetai apì thn epìmenh Prìtash.

Prìtash 4.3.2. 'Estw f : A → R kai èstw x0 ∈ A shmeÐo suss¸reushc touA. Tìte, h f eÐnai suneq c sto x0 an kai mìno an limx→x0 f(x) = f(x0).

Apìdeixh. Upojètoume pr¸ta ìti h f eÐnai suneq c sto x0. 'Estw ε > 0.Up�rqei δ > 0 ¸ste: an x ∈ A kai |x − x0| < δ, tìte |f(x) − f(x0)| < ε.Eidikìtera, an x ∈ A kai 0 < |x − x0| < δ, èqoume |f(x) − f(x0)| < ε. 'Ara,limx→x0 f(x) = f(x0).

AntÐstrofa, ac upojèsoume ìti limx→x0 f(x) = f(x0). 'Estw ε > 0. Up�rqeiδ > 0 ¸ste: an x ∈ A kai 0 < |x−x0| < δ, tìte |f(x)−f(x0)| < ε. Parathr steìti, gia x = x0, èqoume oÔtwc �llwc |f(x) − f(x0)| = 0 < ε. 'Ara, gia k�jex ∈ A me |x− x0| < δ, èqoume |f(x)− f(x0)| < ε. 'Epetai ìti h f eÐnai suneq csto x0. 2

Parat rhsh 4.3.3 (eÐdh asunèqeiac). Ac exet�soume pio prosektik� tishmaÐnei h fr�sh: {h f den eÐnai suneq c sto x0}, ìpou x0 eÐnai shmeÐo sto pedÐoorismoÔ thc f : A → R. Up�rqoun trÐa endeqìmena:

(i) To x0 eÐnai shmeÐo suss¸reushc tou A apì arister� kai apì dexi�, tapleurik� oria thc f kaj¸c x → x0 up�rqoun kai limx→x−0

f(x) = ` =limx→x+

0f(x), ìmwc h tim thc f sto x0 den eÐnai o `: dhlad , f(x0) 6= `.

Tìte lème ìti sto x0 parousi�zetai �rsimh asunèqeia ( {epousi¸dhc}asunèqeia). H f sumperifèretai �rista gÔrw apì to x0, all� h tim thcsto x0 eÐnai {lanjasmènh}.

(ii) To x0 eÐnai shmeÐo suss¸reushc tou A apì arister� kai apì dexi�, tapleurik� ìria thc f kaj¸c x → x0 up�rqoun all� eÐnai diaforetik�:

limx→x−0

f(x) 6= limx→x+

0

f(x).

Tìte lème ìti sto x0 parousi�zetai {asunèqeia a' eÐdouc} ( �lma). Hdiafor� limx→x+

0f(x)− limx→x−0

f(x) eÐnai to {�lma} thc f sto x0.

(iii) To x0 eÐnai shmeÐo suss¸reushc tou A all� to limx→x0 f(x) den up�rqei(gia par�deigma, k�poio apì ta pleurik� ìria thc f kaj¸c x → x0 den up�r-qei). Tìte, lème ìti sto x0 parousi�zetai asunèqeia b' eÐdouc ( {ousi¸dhcasunèqeia}.)

Parat rhsh 4.3.4 (asunèqeiec monìtonwn sunart sewn). 'EstwI èna di�sthma kai èstw f : I → R mia monìtonh sun�rthsh. Tìte ta pleurik�ìria thc f up�rqoun se k�je x0 ∈ I. Sunep¸c, an h f eÐnai asuneq c se k�poiox0 ∈ I, tìte ja parousi�zei �lma sto x0.

Apìdeixh. Upojètoume ìti h f eÐnai aÔxousa kai ìti x0 eÐnai èna eswterikì shmeÐotou I. OrÐzoume

A−(x0) = {f(x) : x ∈ I, x < x0}.To A−(x0) eÐnai mh kenì kai �nw fragmèno apì to f(x0). Sunep¸c, orÐzetaio `− = sup A−(x0). Apì ton orismì tou supremum èqoume `− ≤ f(x0). JadeÐxoume ìti limx→x−0

f(x) = `−.'Estw ε > 0. AfoÔ o `− − ε den eÐnai �nw fr�gma tou sunìlou A−(x0),

up�rqei x < x0 sto I me f(x) > `− − ε. Jètoume δ = x0 − x. Tìte, gia k�jey ∈ (x0 − δ, x0) èqoume y ∈ I (diìti to I eÐnai di�sthma) kai

`− − ε < f(x0 − δ) ≤ f(y) ≤ `− < `− + ε,

diìti h f eÐnai aÔxousa. Dhlad , an y ∈ (x0 − δ, x0), tìte

|f(y)− `−| < ε.

Autì apodeiknÔei ìti limx→x−0f(x) = `− ≤ f(x0), kai me ton Ðdio trìpo deÐqnoume

ìti up�rqei to limx→x+0

f(x) = `+ ≥ f(x0), ìpou

`+ = inf({f(x) : x ∈ I, x > x0}

).

An ta dÔo pleurik� ìria diafèroun, tìte èqoume asunèqeia a' eÐdouc (�lma), en¸an eÐnai Ðsa, tìte f(x0) = limx→x−0

f(x) = limx→x+0

f(x), opìte h f eÐnai suneq csto x0.

4.4 Trigwnometrikèc sunart seic

Se aut th sÔntomh par�grafo upenjumÐzoume k�poiec basikèc tautìthtec kaianisìthtec gia tic trigwnometrikèc sunart seic sin (hmÐtono), cos (sunhmÐtono)kai tan (efaptomènh). 'Ola ta parak�tw eÐnai gnwst� apì to LÔkeio. Austhrìcorismìc twn trigwnometrik¸n ja dojeÐ se epìmeno Kef�laio (jumhjeÐte ìti denèqoume orÐsei akìma ton arijmì π!).

Prìtash 4.4.1. Gia k�je x ∈ R isqÔoun oi

| sin x| ≤ 1, | cosx| ≤ 1 kai sin2 x + cos2 x = 1

kaisin

(π

2− x

)= cos x, cos

(π

2− x

)= sin x.

Oi sunart seic sin : R → [−1, 1] kai cos : R → [−1, 1] eÐnai periodikèc, meel�qisth perÐodo 2π. H sin eÐnai peritt sun�rthsh, en¸ h cos eÐnai �rtia.

4.4 Trigwnometrikec sunarthseic · 99

Prìtash 4.4.2. An 0 < x < π2 , tìte

sin x < x < tanx :=sin x

cosx.

'Epetai ìti, gia k�je x ∈ (−π/2, π/2) isqÔoun oi anisìthtec

| sin x| ≤ |x| ≤ | tanx|

kai ìti gia k�je x ∈ R isqÔei h

| sin x| ≤ |x|.

Prìtash 4.4.3 (sunhmÐtono kai hmÐtono ajroÐsmatoc kai diafo-r�c). Gia k�je a, b ∈ R isqÔoun oi tautìthtec

cos(a− b) = cos a cos b + sin a sin b

cos(a + b) = cos a cos b− sin a sin b

sin(a + b) = sin a cos b + cos a sin b

sin(a− b) = sin a cos b− cos a sin b.

Prìtash 4.4.4 (sunhmÐtono kai hmÐtono tou 2a). Gia k�je a ∈ RisqÔoun oi tautìthtec

cos(2a) = cos2 a− sin2 a = 2 cos2 a− 1 = 1− 2 sin2 a

sin(2a) = 2 sin a cos a.

Prìtash 4.4.5 (metasqhmatismìc ajroÐsmatoc se ginìmeno). Giak�je x, y ∈ R isqÔoun oi tautìthtec

sin x + sin y = 2 sinx + y

2cos

x− y

2

sin x− sin y = 2 sinx− y

2cos

x + y

2

cosx + cos y = 2 cosx + y

2cos

x− y

2

cosx− cos y = 2 sinx + y

2sin

y − x

2.

Prìtash 4.4.6. H sun�rthsh sin : R→ [−1, 1] eÐnai suneq c.

Apìdeixh. 'Estw x0 ∈ R. Gia k�je x ∈ R èqoume

| sin x− sin x0| = 2∣∣∣∣sin

x− x0

2

∣∣∣∣ ·∣∣∣∣cos

x + x0

2

∣∣∣∣ ≤ 2∣∣∣∣sin

x− x0

2

∣∣∣∣ .

Apì thn Prìtash 4.4.2 èqoume∣∣∣∣sin

x− x0

2

∣∣∣∣ ≤∣∣∣∣x− x0

2

∣∣∣∣ .

Sunep¸c,

| sin x− sin x0| ≤ 2∣∣∣∣x− x0

2

∣∣∣∣ = |x− x0|.

T¸ra, eÐnai eÔkolo na doÔme ìti h sin eÐnai suneq c sto x0 (p�rte δ = ε kaiepalhjeÔste ton orismì thc sunèqeiac). H cos eÐnai suneq c wc sÔnjesh thcsuneqoÔc x 7→ π

2 − x me thn sin. 2

Prìtash 4.4.7 (basikì ìrio).

limx→0

sin x

x= 1.

Apìdeixh. H sun�rthsh x 7→ sin xx eÐnai �rtia sto R \ {0}. ArkeÐ loipìn na

deÐxoume (exhg ste giatÐ) ìti

limx→0+

sin x

x= 1.

Apì thn Prìtash 4.4.2 èqoume sin x < x < tanx sto(0, π

2

). Sunep¸c,

cos x <sin x

x< 1

sto(0, π

2

). AfoÔ h cos eÐnai suneq c, èqoume lim

x→0+cos x = cos 0 = 1. Apì to

krit rio parembol c èpetai to zhtoÔmeno. 2

Prìtash 4.4.8. Ta ìria limx→0

sin 1x kai lim

x→0cos 1

x den up�rqoun.

Apìdeixh. Apì thn arq thc metafor�c, arkeÐ na broÔme dÔo akoloujÐec xn → 0,yn → 0 (me xn, yn 6= 0) ¸ste lim sin 1

xn6= lim sin 1

yn. JewroÔme tic akoloujÐec

xn = 1πn kai yn = 1

2πn+ π2

(n ∈ N). EÔkola elègqoume ìti limn xn = 0 = limn yn.'Omwc,

sin1xn

= sin(πn) = 0 → 0

kaisin

1yn

= sin(2πn +

π

2

)= 1 → 1.

TeleÐwc an�loga, mporeÐte na deÐxete ìti to ìrio limx→0

cos 1x den up�rqei. 2

4.5 Ask seic gia ta Kef�laia 3 kai 4



1. An h f : R → R eÐnai suneq c sto x0 kai f(x0) = 1, tìte up�rqei δ > 0¸ste: gia k�je x ∈ (x0 − δ, x0 + δ) isqÔei f(x) > 4

5 .

2. H f : N→ R me f(x) = 1x eÐnai suneq c.

4. H sun�rthsh f : R→ R pou orÐzetai apì tic: f(x) = 0 an x ∈ N kai f(x) = 1an x /∈ N, eÐnai suneq c sto x0 an kai mìno an x0 /∈ N.

4.5 Askhseic gia ta Kefalaia 3 kai 4 · 101

5. Up�rqei f : R → R pou eÐnai asuneq c sta shmeÐa 0, 1, 12 , . . . , 1

n , . . . kaisuneq c se ìla ta �lla shmeÐa.

6. Up�rqei f : R→ R pou eÐnai asuneq c sta shmeÐa 1, 12 , . . . , 1

n , . . . kai suneq cse ìla ta �lla shmeÐa.

7. Up�rqei sun�rthsh f : R→ R pou eÐnai suneq c sto 0 kai asuneq c se ìlata �lla shmeÐa.

8. An h f : R→ R eÐnai suneq c se k�je �rrhto x, tìte eÐnai suneq c se k�jex.

9. An h f eÐnai suneq c sto (a, b) kai f(q) = 0 gia k�je rhtì q ∈ (a, b), tìtef(x) = 0 gia k�je x ∈ (a, b).

10. An h f eÐnai suneq c kai an (xn) eÐnai mia akoloujÐa Cauchy sto pedÐoorismoÔ thc f , tìte h (f(xn)) eÐnai akoloujÐa Cauchy.

11. An f(

1n

)= (−1)n gia k�je n ∈ N, tìte h f eÐnai asuneq c sto shmeÐo 0.

12. An h f : R → R eÐnai suneq c kai f(0) = −f(1) tìte up�rqei x0 ∈ [0, 1]¸ste f(x0) = 0.

13. An h f : (a, b) → R eÐnai suneq c, tìte h f paÐrnei mègisth kai el�qisthtim sto (a, b).

14. An h f eÐnai suneq c sto [a, b] tìte h f eÐnai fragmènh sto [a, b].

15. An limx→0

g(x) = 0 tìte limx→0

g(x) sin 1x = 0.

16. 'Estw f : (a,+∞) → R. An limn→∞

f (a + tn) = L gia k�je gnhsÐwc fjÐnousaakoloujÐa (tn) me tn → 0, tìte lim

x→a+f(x) = L.

17. 'Estw f : (a,+∞) → R. An limn→∞

f(a + 1

n

)= L tìte lim

x→a+f(x) = L.


1. Exet�ste an eÐnai suneqeÐc oi akìloujec sunart seic:

(a) f : R→ R me f(x) =

{x2+x−6

x−2 an x 6= 20 an x = 2

(b) f : R→ R me f(x) ={

sin xx an x 6= 00 an x = 0

(g) fk : [−1, 0] → R me fk(x) ={

xk sin 1x an x 6= 0

0 an x = 0(k ∈ Z)

(d) f : R→ R me f(x) ={

1x sin 1

x2 an x 6= 00 an x = 0

(e) f : R+ → R me f(x) =√

x.

2. DeÐxte ìti h sun�rthsh f : R → R me f(x) =

x an x ∈ Q

x3 an x /∈ QeÐnai

suneq c mìno sta shmeÐa −1, 0, 1.

3. 'Estw f : X → R kai èstw x0 ∈ X. An h f eÐnai suneq c sto x0 kaif(x0) 6= 0, deÐxte ìti:

(a) an f(x0) > 0, up�rqei δ > 0 ¸ste: an |x − x0| < δ kai x ∈ X tìtef(x) > f(x0)

2 > 0.

(b) an f(x0) < 0, up�rqei δ > 0 ¸ste: an |x − x0| < δ kai x ∈ X tìtef(x) < f(x0)

2 < 0.

4. An α ∈ R, h sun�rthsh f : R → R me f(x) = αx profan¸c ikanopoieÐ thnf(x + y) = f(x) + f(y) gia k�je x, y ∈ R.

AntÐstrofa, deÐxte ìti an f : R→ R eÐnai mia suneq c sun�rthsh me f(1) =α, h opoÐa ikanopoieÐ thn f(x + y) = f(x) + f(y) gia k�je x, y ∈ R, tìte:

(a) f(n) = nα gia k�je n ∈ N.

(b) f( 1m ) = α

m gia k�je m = 1, 2, . . ..

(g) f(x) = αx gia k�je x ∈ R.

5. 'Estw f, g : R→ R suneqeÐc sunart seic. DeÐxte ìti:

(a) An f(x) = 0 gia k�je x ∈ Q, tìte f(y) = 0 gia k�je y ∈ R.

(b) An f(x) = g(x) gia k�je x ∈ Q, tìte f(y) = g(y) gia k�je y ∈ R.

(g) An f(x) ≤ g(x) gia k�je x ∈ Q, tìte f(y) ≤ g(y) gia k�je y ∈ R.

6. 'Estw f : R → R suneq c sun�rthsh me f( m2n ) = 0 gia k�je m ∈ Z kai

n ∈ N. DeÐxte ìti f(x) = 0 gia k�je x ∈ R.7. 'Estw f : X → R sun�rthsh. Upojètoume ìti up�rqei M ≥ 0 ¸ste |f(x)−f(y)| ≤ M · |x− y|, gia k�je x ∈ X kai y ∈ X. DeÐxte ìti h f eÐnai suneq c.

8. 'Estw f : R→ R sun�rthsh me |f(x)| ≤ |x| gia k�je x ∈ R.

(a) DeÐxte ìti h f eÐnai suneq c sto 0.

(b) D¸ste par�deigma miac tètoiac f pou na eÐnai asuneq c se k�je x 6= 0.

(g) An g : R→ R eÐnai mia suneq c sun�rthsh me g(0) = 0 kai |f(x)| ≤ |g(x)|gia k�je x ∈ R, tìte h f eÐnai suneq c sto 0.

9. 'Estw f : [a, b] → R suneq c se k�je shmeÐo tou [a, b]. Upojètoume ìtigia k�je x ∈ [a, b] up�rqei y ∈ [a, b] ¸ste |f(y)| ≤ 1

2 |f(x)|. DeÐxte ìti up�rqeix0 ∈ [a, b] ¸ste f(x0) = 0.

10. 'Estw P (x) = amxm + · · ·+ a1x + a0 polu¸numo me thn idiìthta a0am < 0.DeÐxte ìti h exÐswsh P (x) = 0 èqei jetik pragmatik rÐza.

11. 'Estw f : [0, 1] → R suneq c sun�rthsh me thn idiìthta f(x) ∈ Q gia k�jex ∈ [0, 1]. DeÐxte ìti h f eÐnai stajer sun�rthsh.

12. 'Estw f, g : [a, b] → R suneqeÐc sunart seic pou ikanopoioÔn thn f2(x) =g2(x) gia k�je x ∈ [a, b]. Upojètoume epÐshc ìti f(x) 6= 0 gia k�je x ∈ [a, b].DeÐxte ìti g ≡ f g ≡ −f sto [a, b].

13. 'Estw f : [0, 2] → R suneq c sun�rthsh me f(0) = f(2). DeÐxte ìtiup�rqoun x, y ∈ [0, 2] me |x− y| = 1 kai f(x) = f(y).

14. (a) 'Estw f : [a, b] → R suneq c sun�rthsh kai x1, x2 ∈ [a, b]. DeÐxte ìtigia k�je t ∈ [0, 1] up�rqei yt ∈ [a, b] ¸ste

f(yt) = tf(x1) + (1− t)f(x2).

(b) 'Estw f : [a, b] → R suneq c sun�rthsh, kai x1, x2, . . . , xn ∈ [a, b]. DeÐxteìti up�rqei y ∈ [a, b] ¸ste

f(y) =f(x1) + f(x2) + · · ·+ f(xn)

n.

15. 'Estw f : [a, b] → [a, b] suneq c sun�rthsh. Na deiqjeÐ ìti up�rqei x ∈ [a, b]me f(x) = x.

16. 'Estw f : R → R suneq c kai fjÐnousa sun�rthsh. DeÐxte ìti h f èqeimonadikì stajerì shmeÐo: up�rqei akrib¸c ènac pragmatikìc arijmìc x0 gia tonopoÐo

f(x0) = x0.

17. 'Estw f, g : [a, b] → R suneqeÐc sunart seic pou ikanopoioÔn thn f(x) >

g(x) gia k�je x ∈ [a, b]. DeÐxte ìti up�rqei ρ > 0 ¸ste f(x) > g(x)+ρ gia k�jex ∈ [a, b].

18. Qrhsimopoi¸ntac ton orismì tou orÐou, deÐxte ìti

limx→0

√1 + x−√1− x

x= 1 kai lim

x→+∞√

x(√

x + a−√x)

=a

2, a ∈ R.

19. Exet�ste an up�rqoun ta parak�tw ìria kai, an nai, upologÐste ta.

(a) limx→2

x3−8x−2 , (b) lim

x→x0[x], (g) lim

x→x0(x− [x]).

20. 'Estw f : R→ R me f(x) ={

x an x rhtìc−x an x �rrhtoc

. DeÐxte ìti limx→0

f(x) =

0 kai ìti an x0 6= 0 tìte den up�rqei to limx→x0

f(x).

21. DeÐxte ìti an a, b > 0 tìte

limx→0+

x

a

[b

x

]=

b

akai lim

x→0+

b

x

[x

a

]= 0.

Ti gÐnetai ìtan x → 0−?

22. 'Estw α, β, γ > 0 kai λ < µ < ν. DeÐxte ìti h exÐswsh

α

x− λ+

β

x− µ+

γ

x− ν= 0

èqei toul�qiston mÐa rÐza se kajèna apì ta diast mata (λ, µ) kai (µ, ν).

23. 'Estw f, g : R → R dÔo sunart seic. Upojètoume ìti up�rqoun talim

x→x0f(x), lim

x→x0g(x).

(a) DeÐxte ìti an f(x) ≤ g(x) gia k�je x ∈ R, tìte limx→x0

f(x) ≤ limx→x0

g(x).

(b) D¸ste èna par�deigma ìpou f(x) < g(x) gia k�je x ∈ R en¸ limx→x0

f(x) =

limx→x0

g(x).

24. 'Estw X ⊂ R, f, g : X → R dÔo sunart seic kai èstw x0 ∈ R èna shmeÐosuss¸reushc tou X. Upojètoume ìti Ôp�rqei δ > 0 ¸ste h f na eÐnai fragmènhsto (x0 − δ, x0 + δ) ∩X kai ìti lim

x→x0g(x) = 0. DeÐxte ìti lim

x→x0f(x)g(x) = 0.

25. (Je¸rhma Bolzano - Weierstrass) K�je fragmèno kai �peiro uposÔnolo twnpragmatik¸n arijm¸n èqei toul�qiston èna shmeÐo suss¸reushc.

26. 'Estw f : R→ R suneq c sun�rthsh me f(x) > 0 gia k�je x ∈ R kai

limx→−∞

f(x) = limx→+∞

f(x) = 0.

DeÐxte ìti h f paÐrnei mègisth tim : up�rqei y ∈ R ¸ste f(y) ≥ f(x) gia k�jex ∈ R.27. 'Estw f : R→ R suneq c sun�rthsh. An lim

x→−∞f(x) = α kai lim

x→+∞f(x) =

α, tìte h f paÐrnei mègisth el�qisth tim .

28. 'Estw f : R→ R suneq c sun�rthsh me limx→−∞

f(x) = −∞ kai limx→+∞

f(x) =

+∞. DeÐxte ìti f(R) = R.

29. 'Estw f : (α, β) → R sun�rthsh gnhsÐwc aÔxousa kai suneq c. DeÐxte ìti

f((α, β)) = ( limx→α+

f(x), limx→β−

f(x)).

G. Ask seic

1. Melet ste wc proc th sunèqeia th sun�rthsh f : [0, 1] → R me

f(x) =

{0 , x /∈ Q x = 01q , x = p

q , p, q ∈ N,MKD(p, q) = 1.


2. 'Estw f : (a, b) → R kai x0 ∈ (a, b). DeÐxte ìti h f eÐnai suneq c sto x0 ankai mìno an gia k�je monìtonh akoloujÐa (xn) shmeÐwn tou (a, b) me xn → x0

isqÔei f(xn) → f(x0).

3. 'Estw f : R → R suneq c sun�rthsh me thn idiìthta f(x) = f(x + 1

n

)gia

k�je x ∈ R kai k�je n ∈ N. DeÐxte ìti h f eÐnai stajer .

4. Upojètoume ìti h f eÐnai suneq c sto [0, 1] kai f(0) = f(1). 'Estw n ∈ N.DeÐxte ìti up�rqei x ∈ [

0, 1− 1n

]¸ste f(x) = f

(x + 1

n

).

5. 'Estw f, g : [0, 1] → [0, 1] suneqeÐc sunart seic. Upojètoume ìti h f eÐnaiaÔxousa kai g ◦ f = f ◦ g. DeÐxte ìti oi f kai g èqoun koinì stajerì shmeÐo:up�rqei y ∈ [0, 1] ¸ste f(y) = y kai g(y) = y. [Upìdeixh: Xèroume ìti up�rqeix1 ∈ [0, 1] me g(x1) = x1. An isqÔei kai h f(x1) = x1, èqoume telei¸sei. An ìqi,jewr ste thn akoloujÐa xn+1 = f(xn), deÐxte ìti eÐnai monìtonh kai ìti ìloi oiìroi thc eÐnai stajer� shmeÐa thc g. To ìrio thc ja eÐnai koinì stajerì shmeÐotwn f kai g (giatÐ?).]

6. 'Estw f : R → R periodik sun�rthsh me perÐodo T > 0. Upojètoume ìtiup�rqei to lim

x→+∞f(x) = b ∈ R. DeÐxte ìti h f eÐnai stajer .

7. 'Estw f : [0, 1] → R suneq c sun�rthsh. Upojètoume ìti up�rqoun xn ∈[0, 1] ¸ste f(xn) → 0. Tìte, up�rqei x0 ∈ [0, 1] ¸ste f(x0) = 0.

8. 'Estw f : R → R suneq c sun�rthsh kai èstw a1 ∈ R. OrÐzoume an+1 =f(an) gia n = 1, 2, . . .. An an → a tìte f(a) = a.

9. 'Estw f : [a, b] → R me thn ex c idiìthta: gia k�je x0 ∈ [a, b] up�rqei tolim

x→x0f(x). Tìte, h f eÐnai fragmènh.

Gia tic epìmenec dÔo ask seic dÐnoume ton ex c orismì: 'Estw f : X → R. Lèmeìti h f èqei topikì mègisto (antÐstoiqa, topikì el�qisto) sto x0 ∈ X an up�rqeiδ > 0 ¸ste gia k�je x ∈ X ∩ (x0 − δ, x0 + δ) isqÔei h anisìthta f(x) ≤ f(x0)(antÐstoiqa, f(x0) ≤ f(x)).

10. 'Estw f : [a, b] → R suneq c. Upojètoume ìti h f den èqei topikì mègisto el�qisto se kanèna shmeÐo tou (a, b). DeÐxte ìti h f eÐnai monìtonh sto (a, b).

11. 'Estw f : [a, b] → R suneq c sun�rthsh. An h f èqei topikì mègisto sedÔo diaforetik� shmeÐa x1, x2 tou [a, b], tìte up�rqei x3 an�mesa sta x1, x2 stoopoÐo h f èqei topikì el�qisto.

Kef�laio 5

Par�gwgoc

5.1 Orismìc thc parag¸gou

Orismìc 5.1.1. 'Estw f : (a, b) → R mia sun�rthsh kai èstw x0 ∈ (a, b).Lème ìti h f eÐnai paragwgÐsimh sto x0 an up�rqei to ìrio

f ′(x0) := limx→x0

f(x)− f(x0)x− x0

.

To ìrio f ′(x0) (an up�rqei) lègetai par�gwgoc thc f sto x0. Jètontach = x− x0 blèpoume ìti, isodÔnama,

f ′(x0) = limh→0

f(x0 + h)− f(x0)h

an to teleutaÐo ìrio up�rqei.

ParadeÐgmata 5.1.2. (a) 'Estw f : R → R me f(x) = c gia k�je x ∈ R. Hf eÐnai paragwgÐsimh se k�je x0 ∈ R kai f ′(x0) = 0. Pr�gmati,

f(x0 + h)− f(x0)h

=c− c

h= 0 → 0

kaj¸c to h → 0.(b) 'Estw f : R → R me f(x) = x gia k�je x ∈ R. H f eÐnai paragwgÐsimh sek�je x0 ∈ R kai f ′(x0) = 1. Pr�gmati,

f(x0 + h)− f(x0)h

=x0 + h− x0

h=

h

h= 1 → 1

kaj¸c to h → 0.(g) 'Estw f : R→ R me f(x) = |x| gia k�je x ∈ R. H f den eÐnai paragwgÐsimhsto 0 (kai eÐnai paragwgÐsimh se k�je x0 6= 0). Pr�gmati,

limh→0+

f(h)− f(0)h

= limh→0+

|h|h

= limh→0+

h

h= lim

h→0+1 = 1,

108 · Paragwgoc

en¸lim

h→0−

f(h)− f(0)h

= limh→0−

|h|h

= limh→0−

−h

h= lim

h→0−(−1) = −1.

AfoÔ ta dÔo pleurik� ìria eÐnai diaforetik�, to limh→0f(h)−f(0)

h den up�rqei.(d) 'Estw f : R → R me f(x) = x2 gia k�je x ∈ R. H f eÐnai paragwgÐsimh sek�je x0 ∈ R kai f ′(x0) = 2x0. Pr�gmati,

f(x0 + h)− f(x0)h

=(x0 + h)2 − x2

0

h=

2x0h + h2

h= 2x0 + h → 2x0

kaj¸c to h → 0.(e) 'Estw f : R→ R me f(x) = sin x gia k�je x ∈ R. H f eÐnai paragwgÐsimh sek�je x0 ∈ R kai f ′(x0) = cos x0. Pr�gmati,

f(x0 + h)− f(x0)h

=sin(x0 + h)− sinx0

h=

1h· 2 sin

h

2cos

(x0 +

h

2

)→ cos x0

kaj¸c to h → 0, afoÔ limh→0sin(h/2)

h/2 = 1 kai limh→0 cos(x0 + h/2) = cos x0.Me an�logo trìpo mporoÔme na deÐxoume ìti h g : R → R me g(x) = cos x eÐnaiparagwgÐsimh se k�je x0 ∈ R kai g′(x0) = − sinx0.

(st) 'Estw f : R→ R me f(x) ={

x2 an x ∈ Q0 an x /∈ Q . H f eÐnai paragwgÐsimh

sto 0: parathroÔme ìti

f(h)− f(0)h

={

h an h ∈ Q0 an h /∈ Q

'Epetai ìti limh→0f(h)−f(0)

h = 0. Dhlad , f ′(0) = 0. Parathr ste ìti h f eÐnaiasuneq c se k�je x0 6= 0 (kai eÐnai suneq c sto 0).

(z) 'Estw f : R → R me f(x) ={

x3 an x ≥ 0x2 an x < 0

. H f eÐnai paragwgÐsimh se

k�je x0 ∈ R. Gia to shmeÐo 0, jewroÔme to

f(h)− f(0)h

={

h2 an h > 0h an h < 0

'Epetai ìti to ìrio limh→0f(h)−f(0)

h up�rqei kai eÐnai Ðso me 0. Dhlad , f ′(0) = 0.EÔkola elègqoume ìti f ′(x0) = 3x2

0 an x0 > 0 kai f ′(x0) = 2x0 an x0 < 0.

Je¸rhma 5.1.3. 'Estw f : (a, b) → R kai èstw x0 ∈ (a, b). An h f eÐnaiparagwgÐsimh sto x0, tìte h f eÐnai suneq c sto x0.

Apìdeixh. Gia x 6= x0 gr�foume

f(x)− f(x0) =f(x)− f(x0)

x− x0· (x− x0).

AfoÔ

limx→x0

f(x)− f(x0)x− x0

= f ′(x0) kai limx→x0

(x− x0) = 0,

sumperaÐnoume ìti limx→x0(f(x) − f(x0)) = 0, kai sunep¸c, limx→x0 f(x) =f(x0). Autì apodeiknÔei ìti h f eÐnai suneq c sto x0. 2

5.2 Kanonec paragwgishc · 109

Parat rhsh 5.1.4. To antÐstrofo den isqÔei: an h f eÐnai suneq c sto x0,tìte den eÐnai aparaÐthta paragwgÐsimh sto x0. Gia par�deigma, h f(x) = |x|eÐnai suneq c sto 0 all� den eÐnai paragwgÐsimh sto 0.

5.2 Kanìnec parag¸gishc

Qrhsimopoi¸ntac tic antÐstoiqec idiìthtec twn orÐwn, mporoÔme na apodeÐxoumetouc basikoÔc {kanìnec parag¸gishc} se sqèsh me tic �lgebrikèc pr�xeic metaxÔsunart sewn.

Je¸rhma 5.2.1. 'Estw f, g : (a, b) → R dÔo sunart seic kai èstw x0 ∈ (a, b).Upojètoume ìti oi f, g eÐnai paragwgÐsimec sto x0. Tìte:(a) H f + g eÐnai paragwgÐsimh sto x0 kai (f + g)′(x0) = f ′(x0) + g′(x0).(b) Gia k�je t ∈ R, h t · f eÐnai paragwgÐsimh sto x0 kai (t · f)′(x0) = t · f ′(x0).(g) H f ·g eÐnai paragwgÐsimh sto x0 kai (f ·g)′(x0) = f ′(x0)g(x0)+f(x0)g′(x0).(d) An g(x) 6= 0 gia k�je x ∈ (a, b), tìte h f

g eÐnai paragwgÐsimh sto x0 kai

(f

g

)′(x0) =

f ′(x0)g(x0)− f(x0)g′(x0)(g(x0))2

.

Apìdeixh. Ac doÔme gia par�deigma thn apìdeixh tou (g): gr�foume

(f · g)(x0 + h)− (f · g)(x0)h

= f(x0 + h)g(x0 + h)− g(x0)

h

+g(x0)f(x0 + h)− f(x0)

h

gia h 6= 0 (kont� sto 0).'Eqoume limh→0

g(x0+h)−g(x0)h = g′(x0) kai limh→0

f(x0+h)−f(x0)h = f ′(x0).

EpÐshc, h f eÐnai paragwgÐsimh, �ra kai suneq c, sto x0. Sunep¸c, limh→0 f(x0+h) = f(x0). Af nontac to h → 0, kai qrhsimopoi¸ntac tic basikèc idiìthtec twnorÐwn, paÐrnoume to zhtoÔmeno.

Gia to (d) arkeÐ na deÐxoume ìti h 1/g eÐnai paragwgÐsimh sto x0 kai èqeipar�gwgo Ðsh me − g′(x0)

g(x0)2(kai na efarmìsoume to (g)). Parathr ste ìti

limh→0

1h

(1

g(x0 + h)− 1

g(x0)

)= lim

h→0

(g(x0)− g(x0 + h)

h· 1g(x0 + h)g(x0)

)

= −g′(x0) · 1g(x0)2

,

ìpou qrhsimopoi same to gegonìc ìti limh→0 g(x0 +h) = g(x0), pou isqÔei logwthc sunèqeiac thc g sto x0. 2

'Amesec sunèpeiec tou prohgoÔmenou jewr matoc eÐnai oi ex c:

(i) K�je poluwnumik sun�rthsh eÐnai paragwgÐsimh se k�je x0 ∈ R.(ii) K�je rht sun�rthsh eÐnai paragwgÐsimh se k�je shmeÐo tou pedÐou orismoÔ

thc.

110 · Paragwgoc

5.2aþ Kanìnac thc alusÐdac

Prìtash 5.2.2 (parat rhsh tou Karajeodwr ). 'Estw f : (a, b) →R kai èstw x0 ∈ (a, b). H f eÐnai paragwgÐsimh sto x0 an kai mìnon an up�rqeisun�rthsh φ : (a, b) → R pou eÐnai suneq c sto x0 kai ikanopoieÐ thn φ(x) =f(x)−f(x0)

x−x0gia k�je x ∈ (a, b)\{x0}. Tìte, f ′(x0) = φ(x0).

Apìdeixh. Upojètoume pr¸ta ìti h f eÐnai paragwgÐsimh sto x0. OrÐzoumeφ : (a, b) → R wc ex c:

φ(x) =

{f(x)−f(x0)

x−x0an x 6= x0

f ′(x0) an x = x0

H φ eÐnai suneq c sto x0: pr�gmati,

limx→x0

φ(x) = limx→x0

f(x)− f(x0)x− x0

= f ′(x0) = φ(x0).

AntÐstrofa, upojètoume ìti up�rqei φ : (a, b) → R ìpwc sthn Prìtash. AfoÔh φ eÐnai suneq c sto x0, èqoume limx→x0 φ(x) = φ(x0). Dhlad , up�rqei to

limx→x0

f(x)− f(x0)x− x0

= limx→x0

φ(x) = φ(x0).

Apì ton orismì thc parag¸gou, h f eÐnai paragwgÐsimh sto x0 kai f ′(x0) =φ(x0). 2

Je¸rhma 5.2.3 (kanìnac thc alusÐdac). 'Estw f : (a, b) → (c, d) kaig : (c, d) → R dÔo sunart seic. An h f eÐnai paragwgÐsimh sto x0 ∈ (a, b) kai hg eÐnai paragwgÐsimh sto f(x0), tìte h g ◦ f eÐnai paragwgÐsimh sto x0 kai

(g ◦ f)′(x0) = g′(f(x0)) · f ′(x0).

Apìdeixh. Jèloume na deÐxoume ìti to ìrio

limx→x0

g(f(x))− g(f(x0)x− x0

up�rqei kai eÐnai Ðso me g′(f(x0)) · f ′(x0). Jètoume y0 = f(x0) ∈ (c, d) kaijewroÔme th sun�rthsh

ψ : (c, d) → R ìpou ψ(y) =

{g(y)−g(y0)

y−y0an y 6= y0

g′(y0) an y = y0

H ψ eÐnai suneq c sto y0, diìti h g eÐnai paragwgÐsimh sto y0.'Estw x ∈ (a, b)\{x0}. An f(x) 6= f(x0), tìte

ψ(f(x)) =g(f(x))− g(f(x0))

f(x)− f(x0),

�ra èqoume

(∗) g(f(x))− g(f(x0))x− x0

= ψ(f(x))f(x)− f(x0)

x− x0.

5.2 Kanonec paragwgishc · 111

An gia to x isqÔei f(x) = f(x0) tìte h (∗) exakoloujeÐ na isqÔei (ta dÔo mèlhmhdenÐzontai). Dhlad , h (∗) isqÔei gia k�je x ∈ (a, b)\{x0}.

ParathroÔme ìti to ìrio limx→x0f(x)−f(x0)

x−x0up�rqei kai isoÔtai me f ′(x0).

EpÐshc, h f eÐnai suneq c sto x0 kai h ψ eÐnai suneq c sto y0 = f(x0), �ra hsÔnjes touc ψ ◦ f eÐnai suneq c sto x0. Sunep¸c, to ìrio limx→x0 ψ(f(x))up�rqei kai isoÔtai me ψ(y0) = g′(y0) = g′(f(x0)). Epistrèfontac sthn (∗) kaipaÐrnontac to ìrio kaj¸c to x → x0, blèpoume ìti

limx→x0

g(f(x))− g(f(x0))x− x0

= ψ(f(x0)) · f ′(x0) = g′(f(x0)) · f ′(x0),

dhlad to zhtoÔmeno. 2

5.2bþ Par�gwgoc antÐstrofhc sun�rthshc.

Je¸rhma 5.2.4. 'Estw f : (a, b) → R gnhsÐwc monìtonh kai suneq c sun�r-thsh. Upojètoume oti h f eÐnai paragwgÐsimh sto x0 ∈ (a, b) kai ìti f ′(x0) 6= 0.Tìte, h f−1 eÐnai paragwgÐsimh sto f(x0) kai

(f−1)′(f(x0)) =1

f ′(x0).

Apìdeixh. QwrÐc periorismì thc genikìthtac mporoÔme na upojèsoume ìti h f

eÐnai gnhsÐwc aÔxousa. H f ′(x0) up�rqei, dhlad

limx→x0

f(x)− f(x0)x− x0

= f ′(x0).

Epiplèon èqoume upojèsei oti f ′(x0) 6= 0, �ra

limx→x0

x− x0

f(x)− f(x0)=

1f ′(x0)

.

'Estw ε > 0. MporoÔme na broÔme δ > 0 ¸ste [x0 − δ, x0 + δ] ⊂ (a, b) kai an0 < |x− x0| < δ tìte

(∗)∣∣∣∣

x− x0

f(x)− f(x0)− 1

f ′(x0)

∣∣∣∣ < ε.

Jètoumey1 = f (x0 − δ) kai y2 = f (x0 + δ) .

Tìte, to (y1, y2) eÐnai èna anoiqtì di�sthma pou perièqei to f(x0), �ra up�rqeiδ1 > 0 ¸ste

(f(x0)− δ1, f(x0) + δ1) ⊆ (y1, y2) = (f (x0 − δ) , f (x0 + δ)) .

'Estw y pou ikanopoieÐ thn 0 < |y − f(x0)| < δ1. Tìte, y = f(x) gia k�poiox ∈ (a, b) me 0 < |x− x0| < δ. 'Ara,

f−1(y)− f−1(f(x0))y − f(x0)

=x− x0

f(x)− f(x0),

112 · Paragwgoc

opìte h (∗) dÐnei ∣∣∣∣f−1(y)− f−1(f(x0))

y − f(x0)− 1

f ′(x0)

∣∣∣∣ < ε.

Afou to ε > 0 tan tuqìn, èpetai oti

limy→f(x0)

f−1(y)− f−1(f(x0))y − f(x0)

=1

f ′(x0).

Dhlad , h f−1 eÐnai paragwgÐsimh sto f(x0) kai (f−1)′(f(x0)) = 1f ′(x0)

. 2

Parat rhsh 5.2.5. An f ′(x0) = 0 tìte h (f−1)′(y0) den up�rqei. Alli¸c,apì ton kanìna thc alusÐdac h par�gwgoc thc sÔnjeshc f−1 ◦ f sto x0 jaup rqe, kai ja eÐqame

(f−1 ◦ f)′(x0) = (f−1)′(f(x0)) · f ′(x0) = 0.

'Omwc, (f−1 ◦ f)(x) = x, �ra (f−1 ◦ f)′(x0) = 1, opìte odhgoÔmaste se �topo.

5.2gþ Par�gwgoi an¸terhc t�xhc

Orismìc 5.2.6. 'Estw f : (a, b) → R paragwgÐsimh se k�je x ∈ (a, b). H pa-r�gwgoc sun�rthsh thc f eÐnai h sun�rthsh f ′ : (a, b) → R me x 7→ f ′(x).An h sun�rthsh f ′ eÐnai paragwgÐsimh sto (a, b), tìte h par�gwgoc sun�rthshthc f ′ orÐzetai sto (a, b), lègetai deÔterh par�gwgoc thc f , kai sumbolÐ-zetai me f ′′.

Epagwgik�, an èqei oristeÐ h n-ost par�gwgoc f (n) : (a, b) → R thc f kaieÐnai paragwgÐsimh sun�rthsh sto (a, b), tìte h par�gwgoc thc f (n), orÐzetai sto(a, b), lègetai h (n+1)�t�xhc par�gwgoc thc f sto (a, b) kai sumbolÐzetaime f (n+1).

Mia sun�rthsh pou èqei par�gwgo t�xhc n lègetai n forèc paragwgÐsimh.Mia sun�rthsh f : (a, b) → R lègetai aperiìrista paragwgÐsimh stoshmeÐo x0 ∈ (a, b) an h f (n)(x0) up�rqei gia k�je n ∈ N.

Par�deigma 5.2.7. K�je poluwnumik sun�rthsh p(x) = amxm+· · ·+a1x+a0 eÐnai aperiìrista paragwgÐsimh se k�je shmeÐo x0 ∈ R. Oi suntelestèc toupoluwnÔmou p {upologÐzontai} apì tic

ak =p(k)(0)

k!, k = 0, 1, . . . , m.

H apìdeixh gÐnetai me diadoqikèc paragwgÐseic kai upologismì thc p(k)(0). Ank > m, tìte h p(k) eÐnai mhdenÐzetai se k�je x0 ∈ R.

5.3 KrÐsima shmeÐa

Skopìc mac stic epìmenec Paragr�fouc eÐnai na apodeÐxoume ta kÔria jewr matatou DiaforikoÔ LogismoÔ kai na doÔme p¸c efarmìzontai sth melèth sunart se-wn pou orÐzontai se k�poio di�sthma I thc pragmatik c eujeÐac. Ja xekin soume

5.3 Krisima shmeia · 113

me k�poia paradeÐgmata pou deÐqnoun ìti h monotonÐa h Ôparxh k�poiou topikoÔakrìtatou miac paragwgÐsimhc sun�rthshc dÐnoun k�poiec plhroforÐec gia thnpar�gwgo. To monadikì ergaleÐo pou ja qrhsimopoi soume eÐnai o orismìc thcparag¸gou.

L mma 5.3.1. 'Estw f : (a, b) → R paragwgÐsimh sun�rthsh. An h f eÐnaiaÔxousa sto (a, b) tìte f ′(x) ≥ 0 gia k�je x ∈ (a, b).

Apìdeixh. 'Estw x ∈ (a, b). Up�rqei δ > 0 ¸ste (x − δ, x + δ) ⊂ (a, b). Anloipìn |h| < δ tìte h f orÐzetai sto x + h.

AfoÔ h f eÐnai paragwgÐsimh sto x, èqoume

(5.3.1) f ′(x) = limh→0+

f(x + h)− f(x)h

= limh→0−

f(x + h)− f(x)h

.

'Estw 0 < h < δ. AfoÔ h f eÐnai aÔxousa sto (a, b) èqoume f(x + h) ≥ f(x).Sunep¸c,

(5.3.2)f(x + h)− f(x)

h≥ 0 �ra f ′(x) = lim

h→0+

f(x + h)− f(x)h

≥ 0.

Parathr ste ìti deÐxame to zhtoÔmeno qwrÐc na koit�xoume ti gÐnetai gia arnhti-kèc timèc tou h (elègxte ìmwc ìti an −δ < h < 0 tìte h klÐsh f(x+h)−f(x)

h eÐnaip�li mh arnhtik , opìte odhgoÔmaste sto Ðdio sumpèrasma). 2

Parat rhsh 5.3.2. To antÐstrofo er¸thma diatup¸netai wc ex c: an f ′(x) ≥0 gia k�je x ∈ (a, b) tìte eÐnai swstì ìti h f eÐnai aÔxousa sto (a, b)? H ap�-nthsh eÐnai {nai}, aut eÐnai mÐa apì tic basikèc sunèpeiec tou jewr matoc mèshctim c (blèpe §5.4). Qrhsimopoi¸ntac mìno ton orismì thc parag¸gou, mporoÔmena deÐxoume k�ti polÔ asjenèstero:

L mma 5.3.3. 'Estw f : (a, b) → R. Upojètoume ìti h f eÐnai paragwgÐsimhsto x0 ∈ (a, b) kai f ′(x0) > 0. Tìte, up�rqei δ > 0 ¸ste (x0− δ, x0 + δ) ⊆ (a, b)kai

(a) f(x) > f(x0) gia k�je x ∈ (x0, x0 + δ).

(b) f(x) < f(x0) gia k�je x ∈ (x0 − δ, x0).

Apìdeixh. 'Eqoume upojèsei ìti limx→x0

f(x)−f(x0)x−x0

= f ′(x0) > 0. Efarmìzontac

ton ε − δ orismì tou orÐou me ε = f ′(0)2 > 0, brÐskoume δ > 0 ¸ste: an 0 <

|x− x0| < δ tìte x ∈ (a, b) kai

(5.3.3)f(x)− f(x0)

x− x0> f ′(x0)− ε =

f ′(x0)2

> 0.

'Epetai ìti:

(a) Gia k�je x ∈ (x0, x0 + δ) èqoume

(5.3.4) f(x)− f(x0) >f ′(x0)

2(x− x0) > 0 → f(x) > f(x0).

114 · Paragwgoc

(b) Gia k�je x ∈ (x0 − δ, x0) èqoume

(5.3.5) f(x)− f(x0) <f ′(x0)

2(x− x0) < 0 → f(x) < f(x0). 2

Parathr ste ìti oi (5.3.4) kai (5.3.5) den deÐqnoun ìti h f eÐnai aÔxousa sto(x0, x0 + δ) sto (x0 − δ, x0). 2

Parat rhsh 5.3.4. An upojèsoume ìti h paragwgÐsimh sun�rthsh f : (a, b) →R eÐnai gnhsÐwc aÔxousa, den mporoÔme na isquristoÔme ìti h f ′ eÐnai gnhsÐwcjetik sto (a, b). Gia par�deigma, h f : R → R me f(x) = x3 eÐnai gnhsÐwcaÔxousa sto R, ìmwc f ′(x) = 3x2, �ra up�rqei shmeÐo sto opoÐo h par�gwgocmhdenÐzetai: f ′(0) = 0. To L mma 5.3.1 mac exasfalÐzei fusik� ìti f ′ ≥ 0 pantoÔsto R.

Orismìc 5.3.5. 'Estw f : I → R kai èstw x0 ∈ I. Lème ìti h f èqei topikìmègisto sto x0 an up�rqei δ > 0 ¸ste:

an x ∈ I kai |x− x0| < δ tìte f(x0) ≥ f(x).

OmoÐwc, lème ìti h f èqei topikì el�qisto sto x0 an up�rqei δ > 0 ¸ste:

an x ∈ I kai |x− x0| < δ tìte f(x0) ≤ f(x).

An h f èqei topikì mègisto topikì el�qisto sto x0 tìte lème ìti h f èqeitopikì akrìtato sto shmeÐo x0.

Autì pou qreiast kame gia thn apìdeixh tou L mmatoc 5.3.1 tan h Ôparxh thcf ′(x) (o orismìc thc parag¸gou) kai to gegonìc ìti (lìgw monotonÐac) h el�-qisth tim thc f sto [x, x + δ) tan h f(x). Epanalamb�nontac loipìn to Ðdioousiastik� epiqeÐrhma paÐrnoume thn akìloujh Prìtash (Fermat).

Je¸rhma 5.3.6 (Fermat). 'Estw f : [a, b] → R. Upojètoume ìti h f èqeitopikì akrìtato se k�poio x0 ∈ (a, b) kai ìti h f eÐnai paragwgÐsimh sto x0. Tìte,

(5.3.6) f ′(x0) = 0.

Apìdeixh. QwrÐc periorismì thc genikìthtac upojètoume oti h f èqei topikìmègisto sto x0. Up�rqei δ > 0 ¸ste (x0−δ, x0+δ) ⊆ (a, b) kai f(x0+h) ≤ f(x0)gia k�je h ∈ (−δ, δ).

An 0 < h < δ tìte

(5.3.7)f(x0 + h)− f(x0)

h≤ 0, �ra lim

h→0+

f(x0 + h)− f(x0)h

≤ 0.

Sunep¸c, f ′(x0) ≤ 0.An −δ < h < 0 tìte

(5.3.8)f(x0 + h)− f(x0)

h≥ 0, �ra lim

h→0−

f(x0 + h)− f(x0)h

≥ 0.

Sunep¸c, f ′(x0) ≥ 0.Apì tic dÔo anisìthtec èpetai ìti f ′(x0) = 0. 2

5.4 Jewrhma meshc timhc · 115

Orismìc 5.3.7. 'Estw f : I → R. 'Ena eswterikì shmeÐo x0 tou I lègetaikrÐsimo shmeÐo gia thn f an f ′(x0) = 0.

Par�deigma 5.3.8. Ta krÐsima shmeÐa miac sun�rthshc eÐnai polÔ qr simaìtan jèloume na broÔme th mègisth thn el�qisth tim thc. 'Estw f : [a, b] → Rsuneq c sun�rthsh. GnwrÐzoume ìti h f paÐrnei mègisth tim max(f) kai el�qisthtim min(f) sto [a, b]. An x0 ∈ [a, b] kai f(x0) = max(f) f(x0) = min(f),tìte anagkastik� sumbaÐnei k�poio apì ta parak�tw:

(i) x0 = a x0 = b (�kro tou diast matoc).

(ii) x0 ∈ (a, b) kai f ′(x0) = 0 (krÐsimo shmeÐo).

(iii) x0 ∈ (a, b) kai h f den eÐnai paragwgÐsimh sto x0.

Dedomènou ìti, sthn pr�xh, to pl joc twn shmeÐwn pou an koun se autèc tic{treic om�dec} eÐnai sqetik� mikrì, mporoÔme me aplì upologismì kai sÔgkrishmerik¸n tim¸n thc sun�rthshc na apant soume sto er¸thma.Par�deigma: Na brejeÐ h mègisth tim thc sun�rthshc f(x) = x3−x sto [−1, 2].H f eÐnai paragwgÐsimh sto (−1, 2), me par�gwgo f ′(x) = 3x2−1. Ta shmeÐa staopoÐa mhdenÐzetai h par�gwgoc eÐnai ta x1 = − 1√

3kai x2 = 1√

3ta opoÐa an koun

sto (−1, 2). 'Ara, ta shmeÐa sta opoÐa mporeÐ na paÐrnei mègisth el�qisth tim h f eÐnai ta �kra tou diast matoc kai ta dÔo krÐsima shmeÐa:

x0 = −1, x1 = − 1√3, x2 =

1√3, x3 = 2.

Oi antÐstoiqec timèc eÐnai:

f(−1) = 0, f(−1/√

3) =2

3√

3, f(1/

√3) = − 2

3√

3, f(2) = 6.

SugkrÐnontac autèc tic tèsseric timèc blèpoume ìti max(f) = f(2) = 6 kaimin(f) = f(1/

√3) = −2/(3

√3). 2

5.4 Je¸rhma Mèshc Tim c

'Estw f : [a, b] → R mia stajer sun�rthsh. Dhlad , up�rqei c ∈ R ¸stef(x) = c gia k�je x ∈ [a, b]. GnwrÐzoume ìti f ′(x) = 0 gia k�je x ∈ (a, b).AntÐstrofa, ac upojèsoume ìti f : [a, b] → R eÐnai mia suneq c sun�rthsh,paragwgÐsimh sto (a, b), me thn idiìthta f ′(x) = 0 gia k�je x ∈ (a, b). EÐnaiswstì ìti h f eÐnai stajer sto [a, b]?

To er¸thma autì eÐnai parìmoiac fÔshc me ekeÐno thc Parat rhshc 5.3.2: anmia paragwgÐsimh sun�rthsh èqei pantoÔ mh arnhtik par�gwgo, eÐnai swstì ìtieÐnai aÔxousa? EÐnai logikì na perimènoume ìti h ap�nthsh eÐnai {nai} sta dÔoaut� erwt mata. SkefteÐte èna kinhtì: f(x) eÐnai h proshmasmènh apìstashapì thn arqik jèsh th qronik stigm x kai f ′(x) eÐnai h taqÔthta th qronik stigm x. An h taqÔthta eÐnai suneq¸c mhdenik , to kinhtì {mènei akÐnhto} kai hapìstash paramènei stajer . An h taqÔthta eÐnai pantoÔ mh arnhtik , to kinhtì

116 · Paragwgoc

{apomakrÔnetai apì thn arqik tou jèsh} kai h apìstash aux�nei me thn p�rodotou qrìnou.

Gia thn austhr ìmwc apìdeixh aut¸n twn dÔo isqurism¸n, ja qreiasteÐ nasundu�soume thn ènnoia thc parag¸gou me ta basik� jewr mata gia suneqeÐcsunart seic se kleist� diast mata. To basikì teqnikì b ma eÐnai h apìdeixh tou{jewr matoc mèshc tim c}.

'Estw f : [a, b] → R suneq c sun�rthsh. Upojètoume oti h f eÐnai paragw-gÐsimh sto (a, b): dhlad , gia k�je x ∈ (a, b) orÐzetai kal� h efaptomènh tougraf matoc thc f sto (x, f(x)). JewroÔme thn eujeÐa (`) pou pern�ei apì tashmeÐa A = (a, f(a)) kai B = (b, f(b)). An th metakin soume par�llhla proc toneautì thc, k�poia apì tic par�llhlec ja ef�ptetai sto gr�fhma thc f se k�poioshmeÐo (x0, f(x0)), x0 ∈ (a, b). H klÐsh thc efaptomènhc ja prèpei na isoÔtai methn klÐsh thc eujeÐac (`). Dhlad ,

(5.4.1) f ′(x0) =f(b)− f(a)

b− a.

Sto pr¸to mèroc aut c thc paragr�fou dÐnoume austhr apìdeixh autoÔ touisqurismoÔ (Je¸rhma Mèshc Tim c). ApodeiknÔoume pr¸ta mia eidik perÐptwsh:to je¸rhma tou Rolle.

Je¸rhma 5.4.1 (Rolle). 'Estw f : [a, b] → R. Upojètoume ìti h f eÐnaisuneq c sto [a, b] kai paragwgÐsimh sto (a, b). Upojètoume epiplèon ìti f(a) =f(b). Tìte, up�rqei x0 ∈ (a, b) ¸ste

(5.4.2) f ′(x0) = 0.

Apìdeixh. Exet�zoume pr¸ta thn perÐptwsh pou h f eÐnai stajer sto [a, b],dhlad f(x) = f(a) = f(b) gia k�je x ∈ [a, b]. Tìte, f ′(x) = 0 gia k�jex ∈ (a, b) kai opoiod pote apì aut� ta x mporeÐ na paÐxei to rìlo tou x0.

'Estw loipìn ìti h f den eÐnai stajer sto [a, b]. Tìte, up�rqei x1 ∈ (a, b)¸ste f(x1) 6= f(a) kai qwrÐc periorismì thc genikìthtac mporoÔme na upojèsoumeìti f(x1) > f(a). H f eÐnai suneq c sto [a, b], �ra paÐrnei mègisth tim : up�rqeix0 ∈ [a, b] ¸ste

(5.4.3) f(x0) = max{f(x) : x ∈ [a, b]} ≥ f(x1) > f(a).

Eidikìtera, x0 6= a, b. Dhlad , to x0 brÐsketai sto anoiktì di�sthma (a, b). Hf èqei (olikì) mègisto sto x0 kai eÐnai paragwgÐsimh sto x0. Apì to Je¸rhma5.3.6 (Fermat) sumperaÐnoume ìti f ′(x0) = 0. 2

To je¸rhma mèshc tim c eÐnai �mesh sunèpeia tou jewr matoc tou Rolle.

Je¸rhma 5.4.2 (je¸rhma mèshc tim c). 'Estw f : [a, b] → R suneq csto [a, b] kai paragwgÐsimh sto (a, b). Tìte, up�rqei x0 ∈ (a, b) ¸ste

(5.4.4) f ′(x0) =f(b)− f(a)

b− a.

5.4 Jewrhma meshc timhc · 117

Apìdeixh. Ja anaqjoÔme sto Je¸rhma tou Rolle wc ex c. JewroÔme th grammik sun�rthsh h : [a, b] → R pou paÐrnei tic Ðdiec timèc me thn f sta shmeÐa a kai b.Dhlad ,

(5.4.5) h(x) = f(a) +f(b)− f(a)

b− a(x− a).

OrÐzoume mia sun�rthsh g : [a, b] → R me

(5.4.6) g(x) = f(x)− h(x) = f(x)− f(a)− f(b)− f(a)b− a

(x− a).

H g eÐnai suneq c sto [a, b], paragwgÐsimh sto (a, b) kai apì ton trìpo epilog cthc h èqoume

(5.4.7) g(a) = f(a)− h(a) = 0 kai g(b) = f(b)− h(b) = 0.

SÔmfwna me to je¸rhma tou Rolle, up�rqei x0 ∈ (a, b) ¸ste g′(x0) = 0. 'Omwc,

(5.4.8) g′(x) = f ′(x)− f(b)− f(a)b− a

sto (a, b). 'Ara, to x0 ikanopoieÐ thn (5.4.4). 2

Parat rhsh 5.4.3. H upìjesh ìti h f eÐnai suneq c sto kleistì di�sthma[a, b] qrhsimopoi jhke sthn apìdeixh kai eÐnai aparaÐthth. Jewr ste, gia par�-deigma, thn f : [0, 1] → R me f(x) = x an 0 ≤ x < 1 kai f(1) = 0. H f eÐnaiparagwgÐsimh (�ra, suneq c) sto (0, 1) kai èqoume f(0) = f(1) = 0. 'Omwc denup�rqei x ∈ (0, 1) pou na ikanopoieÐ thn f ′(x) = f(1)−f(0)

1−0 = 0, afoÔ f ′(x) = 1gia k�je x ∈ (0, 1). To prìblhma eÐnai sto shmeÐo 1: h f eÐnai asuneq c sto 1,dhlad den eÐnai suneq c sto [0, 1].

To je¸rhma mèshc tim c mac epitrèpei na apant soume sta erwt mata pou suzh-t same sthn arq thc paragr�fou.

Je¸rhma 5.4.4. 'Estw f : (a, b) → R paragwgÐsimh sun�rthsh.

(i) An f ′(x) ≥ 0 gia k�je x ∈ (a, b), tìte h f eÐnai aÔxousa sto (a, b).

(ii) An f ′(x) > 0 gia k�je x ∈ (a, b), tìte h f eÐnai gnhsÐwc aÔxousa sto (a, b).

(iii) An f ′(x) ≤ 0 gia k�je x ∈ (a, b), tìte h f eÐnai fjÐnousa sto (a, b).

(iv) An f ′(x) < 0 gia k�je x ∈ (a, b), tìte h f eÐnai gnhsÐwc fjÐnousa sto(a, b).

(v) An f ′(x) = 0 gia k�je x ∈ (a, b), tìte h f eÐnai stajer sto (a, b).

Apìdeixh. Ja deÐxoume ènan apì touc pr¸touc tèsseric isqurismoÔc: upojètoumeìti f ′(x) ≥ 0 sto (a, b), kai ja deÐxoume ìti an a < x < y < b tìte f(x) ≤ f(y).JewroÔme ton periorismì thc f sto [x, y]. H f eÐnai suneq c sto [x, y] kai

118 · Paragwgoc

paragwgÐsimh sto (x, y), opìte efarmìzontac to je¸rhma mèshc tim c brÐskoumeξ ∈ (x, y) pou ikanopoieÐ thn

(5.4.9) f ′(ξ) =f(y)− f(x)

y − x.

AfoÔ f ′(ξ) ≥ 0 kai y − x > 0, èqoume f(y)− f(x) ≥ 0. Dhlad , f(x) ≤ f(y).Gia ton teleutaÐo isqurismì parathr ste ìti an f ′ = 0 sto (a, b) tìte f ′ ≥ 0

kai f ′ ≤ 0 sto (a, b). 'Ara, h f eÐnai tautìqrona aÔxousa kai fjÐnousa: an x < y

sto (a, b) tìte f(x) ≤ f(y) kai f(x) ≥ f(y), dhlad f(x) = f(y). 'Epetai ìti hf eÐnai stajer . 2

Mia parallag (kai genÐkeush) tou jewr matoc Mèshc Tim c eÐnai to je¸rhmamèshc tim c tou Cauchy:

Je¸rhma 5.4.5 (je¸rhma mèshc tim c tou Cauchy). 'Estw f, g :[a, b] → R, suneqeÐc sto [a, b] kai paragwgÐsimec sto (a, b). Tìte, up�rqei x0 ∈(a, b) ¸ste

(5.4.10) [f(b)− f(a)] g′(x0) = [g(b)− g(a)] f ′(x0).

ShmeÐwsh: Parathr ste pr¸ta ìti to je¸rhma mèshc tim c eÐnai eidik perÐptwshtou jewr matoc pou jèloume na deÐxoume: an g(x) = x tìte g′(x) = 1 kai h(5.4.10) paÐrnei th morf

(5.4.11) [f(b)− f(a)] · 1 = (b− a) f ′(x).

H Ôparxh k�poiou x0 ∈ (a, b) to opoÐo ikanopoieÐ thn (5.4.11) eÐnai akrib¸c oisqurismìc tou jewr matoc mèshc tim c.

JumhjeÐte t¸ra thn idèa thc apìdeixhc tou jewr matoc mèshc tim c. Efar-mìsame to je¸rhma tou Rolle gia th sun�rthsh

(5.4.12) f(x)− f(a)− f(b)− f(a)b− a

(x− a).

IsodÔnama (pollaplasi�ste thn prohgoÔmenh sun�rthsh me b−a) ja mporoÔsamena èqoume p�rei thn

(5.4.13) [f(x)− f(a)] (b− a)− [f(b)− f(a)] (x− a).

Ja jewr soume loipìn sun�rthsh antÐstoiqh me aut n thc (5.4.13) {antikaji-st¸ntac thn x me thn g(x)}.

Apìdeixh. JewroÔme th sun�rthsh h : [a, b] → R me

(5.4.14) h(x) = [f(x)− f(a)] (g(b)− g(a))− [f(b)− f(a)] (g(x)− g(a)).

H h eÐnai suneq c sto [a, b] kai paragwgÐsimh sto (a, b) (giatÐ oi f kai g èqountic Ðdiec idiìthtec). EÔkola elègqoume ìti

(5.4.15) h(a) = 0 = h(b).

5.5 Aprosdioristec morfec · 119

MporoÔme loipìn na efarmìsoume to je¸rhma tou Rolle: up�rqei x0 ∈ (a, b)¸ste h′(x0) = 0. AfoÔ

(5.4.16) h′(x0) = f ′(x0) (g(b)− g(a))− g′(x0) (f(b)− f(a)),

paÐrnoume thn (5.4.10). 2

Parat rhsh 5.4.6. To endiafèron shmeÐo sthn (5.4.10) eÐnai ìti oi par�gw-goi f ′(x0) kai g′(x0) {upologÐzontai sto Ðdio shmeÐo} x0.

PolÔ suqn�, to je¸rhma mèshc tim c tou Cauchy diatup¸netai wc ex c.

Pìrisma 5.4.7. 'Estw f, g : [a, b] → R, suneqeÐc sto [a, b] kai paragwgÐsimecsto (a, b). Upojètoume epiplèon ìti

(a) oi f ′ kai g′ den èqoun koin rÐza sto (a, b).(b) g(b)− g(a) 6= 0.

Tìte up�rqei x0 ∈ (a, b) ¸ste

(5.4.17)f(b)− f(a)g(b)− g(a)

=f ′(x0)g′(x0)

.

Apìdeixh. Apì to je¸rhma mèshc tim c tou Cauchy, up�rqei x0 ∈ (a, b) ¸ste

(5.4.18) (f(b)− f(a))g′(x0) = (g(b)− g(a))f ′(x0).

ParathroÔme ìti g′(x0) 6= 0: an eÐqame g′(x0) = 0, tìte ja tan (g(b) −g(a))f ′(x0) = 0 kai, afoÔ apì thn upìjes mac g(b) − g(a) 6= 0, ja èprepena èqoume f ′(x0) = 0. Dhlad oi f ′ kai g′ ja eÐqan koin rÐza. MporoÔme loipìnna diairèsoume ta dÔo mèlh thc (5.4.18) me (g(b)− g(a))g′(x0) kai na p�roume tozhtoÔmeno. 2

5.5 Aprosdiìristec morfèc

To je¸rhma mèshc tim c tou Cauchy qrhsimopoieÐtai sthn apìdeixh twn {ka-nìnwn tou L’ Hospital} gia ìria thc morf c 0

0 ∞∞ . Tupik� paradeÐgmata thc

kat�stashc pou ja suzht soume se aut thn par�grafo eÐnai ta ex c: jèloumena exet�soume an up�rqei to ìrio

(5.5.1) limx→x0

f(x)g(x)

ìpou f, g eÐnai dÔo sunart seic paragwgÐsimec dexi� kai arister� apì to x0, meg(x) 6= 0 an x kont� sto x0 kai x 6= x0 kai

(5.5.2) limx→x0

f(x) = limx→x0

g(x) = 0

(5.5.3) limx→x0

f(x) = limx→x0

g(x) = +∞.

120 · Paragwgoc

Tìte lème ìti èqoume aprosdiìristh morf 00 ( ∞

∞ antÐstoiqa) sto x0.Oi kanìnec tou l’Hospital mac epitrèpoun suqn� na broÔme tètoia ìria (an

up�rqoun) me th bo jeia twn parag¸gwn twn f kai g. Tupikì je¸rhma autoÔtou eÐdouc eÐnai to ex c.

Je¸rhma 5.5.1. 'Estw f, g : (a, x0)∪ (x0, b) → R paragwgÐsimec sunart seicme tic ex c idiìthtec:

(a) g(x) 6= 0 kai g′(x) 6= 0 gia k�je x ∈ (a, x0) ∪ (x0, b).(b) lim

x→x0f(x) = lim

x→x0g(x) = 0.

An up�rqei to limx→x0

f ′(x)g′(x) = ` ∈ R, tìte up�rqei to lim

x→x0

f(x)g(x) kai

(5.5.4) limx→x0

f(x)g(x)

= ` = limx→x0

f ′(x)g′(x)

.

Apìdeixh. OrÐzoume tic f kai g sto x0 jètontac f(x0) = g(x0) = 0. AfoÔ

(5.5.5) limx→x0

f(x) = limx→x0

g(x) = 0,

oi f kai g gÐnontai t¸ra suneqeÐc sto (a, b). Ja deÐxoume ìti

(5.5.6) limx→x+

0

f(x)g(x)

= ` = limx→x+

0

f ′(x)g′(x)

.

'Eqoume

(5.5.7)f(x)g(x)

=f(x)− f(x0)g(x)− g(x0)

gia k�je x ∈ (x0, b). Oi f ′, g′ den èqoun koin rÐza sto (x0, x) giatÐ h g′ denmhdenÐzetai poujen�. EpÐshc g(x) 6= 0, dhlad g(x) − g(x0) 6= 0. Efarmìzontacloipìn to je¸rhma mèshc tim c tou Cauchy mporoÔme gia k�je x ∈ (x0, b) nabroÔme ξx ∈ (x0, x) ¸ste

(5.5.8)f(x)g(x)

=f ′(ξx)g′(ξx)

.

'Estw t¸ra ìti limx→x0f ′(x)g′(x) = ` kai èstw ε > 0. MporoÔme na broÔme δ > 0

¸ste: an x0 < y < x0 + δ tìte

(5.5.9)∣∣∣∣f ′(y)g′(y)

− `

∣∣∣∣ < ε.

Sundu�zontac tic (5.5.8) kai (5.5.9) blèpoume ìti an x0 < x < x0 + δ tìte

(5.5.10)∣∣∣∣f(x)g(x)

− `

∣∣∣∣ =∣∣∣∣f ′(ξx)g′(ξx)

− `

∣∣∣∣ < ε

(giatÐ x0 < ξx < x < x0 + δ). 'Ara,

(5.5.11) limx→x+

0

f(x)g(x)

= ` = limx→x+

0

f ′(x)g′(x)

.

Me an�logo trìpo deÐqnoume ìti limx→x−0f(x)g(x) = `. 2

O antÐstoiqoc kanìnac ìtan x0 = +∞ eÐnai o ex c.

5.5 Aprosdioristec morfec · 121

Je¸rhma 5.5.2. 'Estw f, g : (a,+∞) → R paragwgÐsimec sunart seic me ticex c idiìthtec:

(a) g(x) 6= 0 kai g′(x) 6= 0 gia k�je x > a.(b) lim

x→+∞f(x) = lim

x→+∞g(x) = 0.

An up�rqei to limx→+∞

f ′(x)g′(x) = ` ∈ R tìte up�rqei to lim

x→+∞f(x)g(x) kai

(5.5.12) limx→+∞

f(x)g(x)

= ` = limx→+∞

f ′(x)g′(x)

.

Apìdeixh. OrÐzoume f1, g1 :(0, 1

a

) → R me

(5.5.13) f1(x) = f

(1x

)kai g1(x) = g

(1x

).

Oi f1, g1 eÐnai paragwgÐsimec kai

(5.5.14)f ′1(x)g′1(x)

=− 1

x2 f ′(

1x

)

− 1x2 g′

(1x

) =f ′

(1x

)

g′(

1x

) .

'Eqoume limx→0+

f1(x) = limx→+∞

f(x) = 0 kai limx→0+

g1(x) = limx→+∞

g(x) = 0 (giatÐ?).

EpÐshc, g1 6= 0 kai g′1 6= 0 sto (0, 1/a). Tèloc,

(5.5.15) limx→0+

f ′1(x)g′1(x)

= limx→0+

f ′(

1x

)

g′(

1x

) = limx→+∞

f ′(x)g′(x)

.

'Ara, efarmìzetai to Je¸rhma 5.5.1 gia tic f1, g1 kai èqoume

(5.5.16) limx→0+

f1(x)g1(x)

= limx→0+

f ′1(x)g′1(x)

= limx→+∞

f ′(x)g′(x)

.

AfoÔ

(5.5.17) limx→+∞

f(x)g(x)

= limx→0+

f1(x)g1(x)

,

èpetai h (5.5.12). 2

Up�rqoun arketèc akìma peript¸seic aprosdiìristwn morf¸n gia tic opoÐecmporoÔme na diatup¸soume kat�llhlo {kanìna tou l’Hospital}. Den ja d¸soume�llec apodeÐxeic, ac doÔme ìmwc th diatÔpwsh enìc kanìna gia aprosdiìristhmorf ∞

∞ .

Je¸rhma 5.5.3. 'Estw f, g : (a, b) → R paragwgÐsimec sunart seic me ticex c idiìthtec:

(a) g(x) 6= 0 kai g′(x) 6= 0 gia k�je x ∈ (a, b).(b) lim

x→a+f(x) = lim

x→a+g(x) = +∞.

An up�rqei to limx→a+

f ′(x)g′(x) = ` ∈ R, tìte up�rqei to lim

x→a+

f(x)g(x) kai

(5.5.18) limx→a+

f(x)g(x)

= limx→a+

f ′(x)g′(x)

.

122 · Paragwgoc

5.6 Idiìthta Darboux gia thn par�gwgo

Orismìc 5.6.1. Lème ìti mia sun�rthsh f : I → R èqei thn idiìthta Darboux(idiìthta thc endi�meshc tim c) an: gia k�je x < y sto I me f(x) 6= f(y) kaigia k�je pragmatikì arijmì ρ an�mesa stouc f(x) kai f(y) mporoÔme na broÔmez ∈ (x, y) ¸ste f(z) = ρ. Apì to Je¸rhma Endi�meshc Tim c èpetai ìti k�jesuneq c sun�rthsh f : I → R èqei thn idiìthta Darboux.

Ja deÐxoume ìti h par�gwgoc miac paragwgÐsimhc sun�rthshc èqei p�nta thnidiìthta Darboux (an kai den eÐnai p�nta suneq c sun�rthsh).

Je¸rhma 5.6.2. 'Estw f : (a, b) → R paragwgÐsimh sun�rthsh. Tìte, h f ′

èqei thn idiìthta Darboux.

Apìdeixh: 'Estw x < y ∈ (a, b) me f ′(x) 6= f ′(y). QwrÐc periorismì thcgenikìthtac mporoÔme na upojèsoume ìti f ′(x) < f ′(y). Upojètoume ìti f ′(x) <

ρ < f ′(y) kai ja broÔme z ∈ (x, y) ¸ste f ′(z) = ρ.JewroÔme th sun�rthsh g : (a, b) → R pou orÐzetai apì thn g(t) = f(t)− ρt.

Tìte, h g eÐnai paragwgÐsimh sto (a, b) kai g′(t) = f ′(t) − ρ. 'Ara, èqoumeg′(x) < 0 < g′(y) kai zht�me z ∈ (x, y) me thn idiìthta g′(z) = 0.

Isqurismìc. Up�rqoun x1, y1 ∈ (x, y) ¸ste g(x1) < g(x) kai g(y1) < g(y).

Apìdeixh tou isqurismoÔ. H g eÐnai paragwgÐsimh sto x, dhlad

(5.6.1) limh→0+

g(x + h)− g(x)h

= g′(x) < 0.

Epilègontac ε = − g′(x)2 > 0 blèpoume ìti up�rqei 0 < δ1 < y − x ¸ste

(5.6.2)g(x + h)− g(x)

h< g′(x) + ε =

g′(x)2

< 0

gia k�je 0 < h < δ1. PaÐrnontac x1 = x+ δ12 èqoume x1 ∈ (x, y) kai g(x1) < g(x).

'Omoia, h g eÐnai paragwgÐsimh sto y, dhlad

(5.6.3) limh→0−

g(y + h)− g(y)h

= g′(y) > 0.

Epilègontac ε = g′(y)2 > 0 blèpoume ìti up�rqei 0 < δ1 < y − x ¸ste

(5.6.4)g(y + h)− g(y)

h> g′(y)− ε =

g′(y)2

> 0

gia k�je −δ1 < h < 0. PaÐrnontac y1 = y − δ12 èqoume y1 ∈ (x, y) kai g(y1) <

g(y). 2

Sunèqeia thc apìdeixhc tou Jewr matoc 5.6.2. H g eÐnai paragwgÐsimh sto (a, b),�ra suneq c sto [x, y]. Epomènwc, h g paÐrnei el�qisth tim sto [x, y]: up�rqeix0 ∈ [x, y] me thn idiìthta g(x0) ≤ g(t) gia k�je t ∈ [x, y].

Apì ton Isqurismì blèpoume ìti h g den paÐrnei el�qisth tim sto x oÔtesto y. 'Ara, x0 ∈ (x, y). AfoÔ h g eÐnai paragwgÐsimh sto x0, to Je¸rhma 5.3.6(Fermat) mac exasfalÐzei ìti g′(x0) = 0. 'Epetai to zhtoÔmeno, me z = x0. 2

5.7 Gewmetrikh shmasia thc deuterhc paragwgou · 123

5.7 Gewmetrik shmasÐa thc deÔterhc parag¸gou

Sthn §5.3 eÐdame ìti o mhdenismìc thc parag¸gou se èna shmeÐo x0 den eÐnaiikan sunj kh gia thn Ôparxh topikoÔ akrìtatou sto x0. H sun�rthsh f(x) =x3 den èqei akrìtato sto x0 = 0, ìmwc f ′(x0) = 0. Koit�zontac th deÔterhpar�gwgo sta shmeÐa mhdenismoÔ thc pr¸thc parag¸gou mporoÔme pollèc forècna sumper�noume an èna krÐsimo shmeÐo eÐnai ìntwc shmeÐo akrìtatou.

Je¸rhma 5.7.1. 'Estw f : (a, b) → R paragwgÐsimh sun�rthsh kai èstwx0 ∈ (a, b) me f ′(x0) = 0.

(a) An up�rqei h f ′′(x0) kai f ′′(x0) > 0, tìte èqoume topikì el�qisto sto x0.

(b) An up�rqei h f ′′(x0) kai f ′′(x0) < 0, tìte èqoume topikì mègisto sto x0.

ShmeÐwsh: An f ′′(x0) = 0 an den up�rqei h f ′′(x0), tìte prèpei na exet�soumeti sumbaÐnei me �llo trìpo.

Apìdeixh. Ja deÐxoume mìno to (a). 'Eqoume

(5.7.1) 0 < f ′′(x0) = limx→x0

f ′(x)− f ′(x0)x− x0

= limx→x0

f ′(x)x− x0

.

Epomènwc, mporoÔme na broÔme δ > 0 ¸ste:

(i) An x0 < x < x0 + δ, tìte f ′(x) > 0.

(ii) An x0 − δ < x < x0 tìte f ′(x) < 0.

'Estw y ∈ (x0 − δ, x0 + δ).

(i) An x0 < y < x0 + δ, tìte efarmìzontac to je¸rhma mèshc tim c sto [x0, y]brÐskoume x ∈ (x0, y) ¸ste

(5.7.2) f(y)− f(x) = f ′(x)(y − x0) > 0.

(ii) An x0− δ < y < x0, tìte efarmìzontac to je¸rhma mèshc tim c sto [y, x0]brÐskoume x ∈ (y, x0) ¸ste

(5.7.3) f(y)− f(x) = f ′(x)(y − x0) > 0.

Dhlad , f(y) ≤ f(x0) gia k�je y ∈ (x0 − δ, x0 + δ). 'Ara, h f èqei topikìel�qisto sto x0. 2

5.7aþ Kurtèc kai koÐlec sunart seic

Se epìmeno Kef�laio ja asqolhjoÔme susthmatik� me tic kurtèc kai tic koÐlecsunart seic f : I → R. Se aut thn upopar�grafo apodeiknÔoume k�poiec aplècprot�seic gia paragwgÐsimec sunart seic, oi opoÐec mac bohj�ne na {sqedi�soumeth grafik touc par�stash}.

124 · Paragwgoc

JewroÔme mia paragwgÐsimh sun�rthsh f : (a, b) → R. An x0 ∈ (a, b), h{exÐswsh thc efaptomènhc} tou graf matoc thc f sto (x0, f(x0)) eÐnai h

(5.7.4) y = f(x0) + f ′(x0)(x− x0).

Lème ìti h f eÐnai kurt sto (a, b) an gia k�je x0 ∈ (a, b) èqoume

(5.7.5) f(x) ≥ f(x0) + f ′(x0)(x− x0)

gia k�je x ∈ (a, b). Dhlad , an to gr�fhma {(x, f(x)) : a < x < b} brÐsketaip�nw apì thn efaptomènh. Lème ìti h f eÐnai gnhsÐwc kurt sto (a, b) an giak�je x 6= x0 h anisìthta sthn (5.7.5) eÐnai gn sia.Lème ìti h f eÐnai koÐlh sto (a, b) an gia k�je x0 ∈ (a, b) èqoume

(5.7.6) f(x) ≤ f(x0) + f ′(x0)(x− x0)

gia k�je x ∈ (a, b). Dhlad , an to gr�fhma {(x, f(x)) : a < x < b} brÐsketaik�tw apì thn efaptomènh. Lème ìti h f eÐnai gnhsÐwc koÐlh sto (a, b) an giak�je x 6= x0 h anisìthta sthn (5.7.6) eÐnai gn sia.Tèloc, lème ìti h f èqei shmeÐo kamp c sto shmeÐo x0 ∈ (a, b) an up�rqei δ > 0¸ste h f na eÐnai gnhsÐwc kurt sto (x0−δ, x0) kai gnhsÐwc koÐlh sto (x0, x0+δ) gnhsÐwc koÐlh sto (x0 − δ, x0) kai gnhsÐwc kurt sto (x0, x0 + δ).

Je¸rhma 5.7.2. 'Estw f : (a, b) → R paragwgÐsimh sun�rthsh.(a) An h f ′ eÐnai (gnhsÐwc) aÔxousa sto (a, b), tìte h f eÐnai (gnhsÐwc) kurt sto (a, b).(b) An h f ′ eÐnai (gnhsÐwc) fjÐnousa sto (a, b), tìte h f eÐnai (gnhsÐwc) koÐlhsto (a, b).

Apìdeixh. 'Estw x0 ∈ (a, b) kai èstw x ∈ (a, b). Upojètoume pr¸ta ìti x > x0.Apì to je¸rhma mèshc tim c, up�rqei ξx ∈ (x0, x) me thn idiìthta

(5.7.7) f(x)− f(x0) = (x− x0)f ′(ξx).

AfoÔ x0 < ξx èqoume f ′(ξx) ≥ f ′(x0), kai afoÔ x− x0 > 0 blèpoume ìti

(5.7.8) f(x)− f(x0) = (x− x0)f ′(ξx) ≥ (x− x0)f ′(x0).

Upojètoume t¸ra ìti x < x0. Apì to je¸rhma mèshc tim c, up�rqei ξx ∈ (x, x0)me thn idiìthta

(5.7.9) f(x)− f(x0) = (x− x0)f ′(ξx).

AfoÔ ξx < x0 èqoume f ′(ξx) ≤ f ′(x0), kai afoÔ x− x0 < 0 blèpoume ìti

(5.7.10) f(x)− f(x0) = (x− x0)f ′(ξx) ≥ (x− x0)f ′(x0).

Se k�je perÐptwsh, isqÔei h (5.7.5). Elègxte ìti an h f ′ upotejeÐ gnhsÐwcaÔxousa sto (a, b) tìte paÐrnoume gn sia anisìthta sthn (5.7.5).(b) Me ton Ðdio trìpo. 2

H deÔterh par�gwgoc (an up�rqei) mporeÐ na mac d¸sei plhroforÐa gia to an hf eÐnai kurt koÐlh.

5.7 Gewmetrikh shmasia thc deuterhc paragwgou · 125

Je¸rhma 5.7.3. 'Estw f : (a, b) → R dÔo forèc paragwgÐsimh sun�rthsh.

(a) An f ′′(x) > 0 gia k�je x ∈ (a, b), tìte h f eÐnai gnhsÐwc kurt sto (a, b).

(b) An f ′′(x) < 0 gia k�je x ∈ (a, b), tìte h f eÐnai gnhsÐwc koÐlh sto (a, b).

Apìdeixh: (a) Afou f ′′ > 0 sto (a, b), h f ′ eÐnai gnhsÐwc aÔxousa sto (a, b)(Je¸rhma 5.4.4). Apì to Je¸rhma 5.7.2 èpetai to zhtoÔmeno.

(b) Me ton Ðdio trìpo. 2

Tèloc, dÐnoume mia anagkaÐa sunj kh gia na eÐnai to x0 shmeÐo kamp c thc f .

Je¸rhma 5.7.4. 'Estw f : (a, b) → R dÔo forèc paragwgÐsimh sun�rthsh kaièstw x0 ∈ (a, b). An h f èqei shmeÐo kamp c sto x0, tìte f ′′(x0) = 0.

Apìdeixh. JewroÔme th sun�rthsh g(x) = f(x) − f(x0) − f ′(x0)(x − x0). H g

den èqei topikì mègisto el�qisto sto x0: èqoume g(x0) = 0 kai g > 0 arister�tou x0, g < 0 dexi� tou x0 � to antÐstrofo.

EpÐshc, g′(x0) = 0 kai g′′(x0) = f ′′(x0). An tan g′′(x0) > 0 g′′(x0) < 0tìte apì to Je¸rhma 5.7.1 h g ja eÐqe akrìtato sto x0, �topo. 'Ara, f ′′(x0) = 0.2

ShmeÐwsh. H sunj kh tou Jewr matoc 5.7.4 den eÐnai ikan . H f(x) = x4 denèqei shmeÐo kamp c sto x0 = 0. EÐnai gnhsÐwc kurt sto R. 'Omwc f ′′(x) = 12x2,�ra f ′′(0) = 0.

Par�deigma. Melet ste th sun�rthsh f : R→ R me

(5.7.11) f(x) =1

x2 + 1.

H f eÐnai paragwgÐsimh sto R, me

(5.7.12) f ′(x) = − 2x

(x2 + 1)2.

'Ara, h f eÐnai gnhsÐwc aÔxousa sto (−∞, 0) kai gnhsÐwc fjÐnousa sto (0, +∞).PaÐrnei mègisth tim sto 0: f(0) = 1, kai lim

x→±∞f(x) = 0. H deÔterh par�gwgoc

thc f orÐzetai pantoÔ kai eÐnai Ðsh me

(5.7.13) f ′′(x) =2(3x2 − 1)(x2 + 1)3

.

'Ara, f ′′ > 0 sta(−∞,− 1√

3

)kai

(1√3,∞

), en¸ f ′′ < 0 sto

(− 1√

3, 1√

3

). 'Epetai

ìti h f èqei shmeÐo kamp c sta ± 1√3kai eÐnai: gnhsÐwc kurt sta

(−∞,− 1√

3

)

kai(

1√3,∞

), gnhsÐwc koÐlh sto

(− 1√

3, 1√

3

). Autèc oi plhroforÐec eÐnai arketèc

gia na sqedi�soume {arket� pist�} th grafik par�stash thc f .

126 · Paragwgoc

5.7bþ AsÔmptwtec

1. 'Estw f : (a, +∞) → R.(a) Lème ìti h eujeÐa y = β eÐnai orizìntia asÔmptwth thc f sto +∞ an

(5.7.14) limx→+∞

f(x) = β.

Par�deigma: h f : (1,+∞) → R me f(x) = x+1x−1 èqei orizìntia asÔmptwth thn

y = 1.(b) Lème ìti h eujeÐa y = αx+ β (α 6= 0) eÐnai pl�gia asÔmptwth thc f sto +∞an

(5.7.15) limx→+∞

(f(x)− (αx + β)) = 0.

Parathr ste ìti h f èqei to polÔ mÐa pl�gia asÔmptwth sto +∞ kai ìti any = αx + β eÐnai h asÔmptwth thc f tìte h klish thc α upologÐzetai apì thn

(5.7.16) α = limx→+∞

f(x)x

kai h stajer� β upologÐzetai apì thn

(5.7.17) β = limx→+∞

(f(x)− αx).

AntÐstrofa, gia na doÔme an h f èqei pl�gia asÔmptwth sto +∞, exet�zoumepr¸ta an up�rqei to lim

x→+∞f(x)

x . An autì to ìrio up�rqei kai an eÐnai diaforetikì

apì to 0, to sumbolÐzoume me α kai exet�zoume an up�rqei to limx→+∞

(f(x)−αx).An kai autì to ìrio � ac to poÔme β � up�rqei, tìte h y = αx + β eÐnai h pl�giaasÔmptwth thc f sto +∞.Par�deigma: h f : (1,+∞) → R me f(x) = x2+x−1

x−1 èqei pl�gia asÔmptwth sto+∞ thn y = x + 2. Pr�gmati,

(5.7.18) limx→+∞

f(x)x

= limx→+∞

x2 + x− 1x2 − x

= 1,

kai

(5.7.19) limx→+∞

(f(x)− x) = limx→+∞

2x− 1x− 1

= 2.

2. Me an�logo trìpo orÐzoume � kai brÐskoume � thn orizìntia pl�gia asÔm-ptwth miac sun�rthshc f : (−∞, a) → R sto −∞ (an up�rqei).

3. Tèloc, lème ìti h f : (a, x0)∪ (x0, b) → R èqei (arister dexi�) katakìrufhasÔmptwth sto x0 an

(5.7.20) limx→x−0

f(x) = ±∞ limx→x+

0

f(x) = ±∞

antÐstoiqa. Gia par�deigma, h f(x) = 1x èqei arister kai dexi� asÔmptwth sto

0 thn eujeÐa x = 0, afoÔ limx→0−

1x = −∞ kai lim

x→0+

1x = +∞.

5.8 Askhseic · 127

5.8 Ask seic


A1. Exet�ste an oi parak�tw prot�seic eÐnai alhjeÐc yeudeÐc (aitiolog stepl rwc thn ap�nthsh sac).

1. An h f eÐnai paragwgÐsimh sto (a, b), tìte h f eÐnai suneq c sto (a, b).

2. An h f eÐnai paragwgÐsimh sto x0 = 0 kai an f(0) = f ′(0) = 0, tìtelim

n→∞nf(1/n) = 0.

3. An h f eÐnai paragwgÐsimh sto [a, b] kai paÐrnei th mègisth tim thc stox0 = a, tìte f ′(a) = 0.

4. An f ′(x) ≥ 0 gia k�je x ∈ [0,∞) kai f(0) = 0, tìte f(x) ≥ 0 gia k�jex ∈ [0,∞).

5. An h f eÐnai dÔo forèc paragwgÐsimh sto [0, 2] kai f(0) = f(1) = f(2) = 0,tìte up�rqei x0 ∈ (0, 2) ¸ste f ′′(x0) = 0.

6. 'Estw f : (a, b) → R kai èstw x0 ∈ (a, b). An h f eÐnai suneq c sto x0,paragwgÐsimh se k�je x ∈ (a, b) \ {x0} kai an up�rqei to lim

x→x0f ′(x) = ` ∈ R,

tìte f ′(x0) = `.

7. An h f : R → R eÐnai paragwgÐsimh sto 0, tìte up�rqei δ > 0 ¸ste h f naeÐnai suneq c sto (−δ, δ).

8*. An h f eÐnai paragwgÐsimh sto x0 ∈ R kai f ′(x0) > 0, tìte up�rqei δ > 0¸ste h f na eÐnai gnhsÐwc aÔxousa sto (x0 − δ, x0 + δ).

A2. D¸ste par�deigma sun�rthshc f : R→ R me tic ex c idiìthtec:

(a) f(−1) = 0, f(2) = 1 kai f ′(1) > 0.

(b) f(−1) = 0, f(2) = 1 kai f ′(1) < 0.

(g) f(0) = 0, f(3) = 1, f ′(1) = 0 kai h f eÐnai gnhsÐwc aÔxousa sto [0, 3].

(d) f(m) = 0 kai f ′(m) = (−1)m gia k�je m ∈ Z, |f(x)| ≤ 12 gia k�je x ∈ R.


1. Gia kajemÐa apì tic parak�tw sunart seic breÐte th mègisth kai thn el�qisthtim thc sto di�sthma pou upodeiknÔetai.

(a) f(x) = x3 − x2 − 8x + 1 sto [−2, 2].

(b) f(x) = x5 + x + 1 sto [−1, 1].

(g) f(x) = x3 − 3x sto [−1, 2].

128 · Paragwgoc

2. DeÐxte ìti h exÐswsh:

(a) 4ax3 + 3bx2 + 2cx = a + b + c èqei toul�qiston mÐa rÐza sto (0, 1).

(b) 6x4 − 7x + 1 = 0 èqei to polÔ dÔo pragmatikèc rÐzec.

(g) x3 + 9x2 + 33x− 8 = 0 èqei akrib¸c mÐa pragmatik rÐza.

3. DeÐxte ìti h exÐswsh xn + ax + b = 0 èqei to polÔ dÔo pragmatikèc rÐzec ano n eÐnai �rtioc kai to polÔ treic pragmatikèc rÐzec an o n eÐnai perittìc.

4. 'Estw a1 < · · · < an sto R kai èstw f(x) = (x− a1) · · · (x− an). DeÐxte ìtih exÐswsh f ′(x) = 0 èqei akrib¸c n− 1 lÔseic.

5. Sqedi�ste tic grafikèc parast�seic twn sunart sewn

f(x) = x +1x

, f(x) = x +3x2

, f(x) =x2

x2 − 1, f(x) =

11 + x2

jewr¸ntac san pedÐo orismoÔ touc to megalÔtero uposÔnolo tou R sto opoÐomporoÔn na oristoÔn.

6. (a) DeÐxte ìti: apì ìla ta orjog¸nia parallhlìgramma me stajer diag¸nio,to tetr�gwno èqei to mègisto embadìn.

(b) DeÐxte ìti: apì ìla ta orjog¸nia parallhlìgramma me stajer perÐmetro,to tetr�gwno èqei to mègisto embadìn.

7. BreÐte ta shmeÐa thc uperbol c x2 − y2 = 1 pou èqoun el�qisth apìstashapì to shmeÐo (0, 1).

8. P�nw se kÔklo aktÐnac 1 jewroÔme dÔo antidiametrik� shmeÐa A, B. BreÐteta shmeÐa Γ tou kÔklou gia ta opoÐa to trÐgwno ABΓ èqei th mègisth dunat perÐmetro.

9. DÐnontai pragmatikoÐ arijmoÐ a1 < a2 < · · · < an. Na brejeÐ h el�qisth tim

thc sun�rthshc f(x) =n∑

k=1

(x− ak)2.

10. 'Estw a > 0. DeÐxte ìti h mègisth tim thc sun�rthshc

f(x) =1

1 + |x| +1

1 + |x− a|

eÐnai Ðsh me 2+a1+a .

11. 'Estw f : (a, b) → R, a < x0 0. DeÐxte ìti up�rqei δ > 0¸ste

(a) f(x) > f(x0) gia k�je x ∈ (x0, x0 + δ).

(b) f(x) < f(x0) gia k�je x ∈ (x0 − δ, x0).

5.8 Askhseic · 129

12. 'Estw f : (a, b) → R paragwgÐsimh sun�rthsh kai èstw x0 ∈ (a, b). Upojè-toume ìti up�rqei h f ′′(x0). DeÐxte ìti:

(a) An h f èqei topikì el�qisto sto x0, tìte f ′′(x0) ≥ 0.

(b) An h f èqei topikì mègisto sto x0, tìte f ′′(x0) ≤ 0.

13. Upojètoume ìti oi sunart seic f kai g eÐnai paragwgÐsimec sto [a, b] kai ìtif(a) = g(a) kai f(b) = g(b). DeÐxte ìti up�rqei toul�qiston èna shmeÐo x sto(a, b) gia to opoÐo oi efaptìmenec twn grafik¸n parast�sewn twn f kai g sta(x, f(x)) kai (x, g(x)) eÐnai par�llhlec tautÐzontai.

14. DÐnontai dÔo paragwgÐsimec sunart seic f, g : (a, b) → R ¸ste f(x)g′(x)−f ′(x)g(x) 6= 0 gia k�je x ∈ (a, b). DeÐxte ìti an�mesa se dÔo rÐzec thc f(x) = 0brÐsketai mia rÐza thc g(x) = 0, kai antÐstrofa.

15. 'Estw f : [a, b] → R, suneq c sto [a, b], paragwgÐsimh sto (a, b), me f(a) =f(b). DeÐxte ìti up�rqoun x1 6= x2 ∈ (a, b) ¸ste f ′(x1) + f ′(x2) = 0.

16. 'Estw f : (0, +∞) → R paragwgÐsimh, me limx→+∞

f ′(x) = 0. DeÐxte ìti

limx→+∞

(f(x + 1)− f(x)) = 0.

17. 'Estw f : (1,+∞) → R paragwgÐsimh sun�rthsh me thn idiìthta: |f ′(x)| ≤1x gia k�je x > 1. DeÐxte ìti lim

x→+∞[f(x +

√x)− f(x)] = 0.

18. 'Estw f, g dÔo sunart seic suneqeÐc sto [0, a] kai paragwgÐsimec sto (0, a).Upojètoume ìti f(0) = g(0) = 0 kai f ′(x) > 0, g′(x) > 0 sto (0, a).

(a) An h f ′ eÐnai aÔxousa sto (0, a), deÐxte ìti h f(x)x eÐnai aÔxousa sto (0, a).

(b) An h f ′

g′ eÐnai aÔxousa sto (0, a), deÐxte ìti h fg eÐnai aÔxousa sto (0, a).

G. Ask seic*

1. DÐnontai pragmatikoÐ arijmoÐ a1 < a2 < · · · < an. Na brejeÐ h el�qisth tim

thc sun�rthshc g(x) =n∑

k=1

|x− ak|.

2. 'Estw f paragwgÐsimh sto (0, 2) me |f ′(x)| ≤ 1 gia k�je x ∈ (0, 2). Jètoumean = f

(1n

). DeÐxte ìti h (an) sugklÐnei.

3. 'Estw n ∈ N kai èstw f(x) = (x2 − 1)n. DeÐxte ìti h exÐswsh f (n)(x) = 0èqei akrib¸c n diaforetikèc lÔseic, ìlec sto di�sthma (−1, 1).

4. 'Estw a > 0. DeÐxte ìti den up�rqei paragwgÐsimh sun�rthsh f : [0, 1] → Rme f ′(0) = 0 kai f ′(x) ≥ a gia k�je x ∈ (0, 1].

130 · Paragwgoc

5. 'Estw f : (a, b) → R paragwgÐsimh sun�rthsh. An h f ′ eÐnai asuneq c sek�poio shmeÐo x0 ∈ (a, b), deÐxte oti h asunèqeia thc f ′ sto x0 eÐnai ousi¸dhc(den up�rqei to ìrio lim

x→x0f ′(x)).

6. 'Estw f : (a, b) → R paragwgÐsimh sun�rthsh me limx→b−

f(x) = +∞. DeÐxte

ìti an up�rqei to limx→b−

f ′(x) tìte eÐnai Ðso me +∞.

7. 'Estw f : (0, +∞) → R paragwgÐsimh sun�rthsh me limx→+∞

f(x) = L ∈ R.DeÐxte ìti an up�rqei to lim

x→+∞f ′(x) tìte eÐnai Ðso me 0.

Kef�laio 6

UpodeÐxeic gia ticAsk seic

6.1 PragmatikoÐ ArijmoÐ


1. Swstì. O supA eÐnai ex orismoÔ �nw fr�gma tou A. Sunep¸c, gia k�jex ∈ A èqoume x ≤ supA.

2. Swstì. An o x eÐnai �nw fr�gma tou A tìte supA ≤ x apì ton orismì tousupA: o supA eÐnai to mikrìtero �nw fr�gma tou A. AntÐstrofa, an supA ≤ x

tìte gia k�je a ∈ A èqoume a ≤ sup A ≤ x, dhlad o x eÐnai �nw fr�gma tou A.

3. L�joc. To A = (0, 1) = {x ∈ R : 0 < x < 1} eÐnai mh kenì kai �nw fragmènouposÔnolo tou R, ìmwc sup A = 1 /∈ A.

4. Swstì. 'Estw A èna mh kenì kai �nw fragmèno uposÔnolo tou Z. Apì toaxÐwma thc plhrìthtac, up�rqei to a = sup A ∈ R. Ja deÐxoume ìti a ∈ A: apìton qarakthrismì tou supremum, up�rqei x ∈ A ¸ste a−1 < x. An a /∈ A, tìtex < a. Autì shmaÐnei ìti o x den eÐnai �nw fr�gma tou A, opìte, efarmìzontacp�li ton qarakthrismì tou supremum, brÐskoume y ∈ A ¸ste a−1 < x < y < a.'Epetai ìti 0 < y − x < 1. Autì eÐnai �topo, diìti oi x kai y eÐnai akèraioi.

5. Swstì. An a = sup A kai ε > 0, apì ton qarakthrismì tou supremum,up�rqei x ∈ A ¸ste a− ε < x. Apì thn �llh pleur�, o a eÐnai �nw fr�gma touA kai x ∈ A. Sunep¸c, x ≤ a.

6. L�joc. P�rte, gia par�deigma, A = {1, 2}. Tìte, supA = 2. An ìmwcp�roume ε = 1

2 , tìte den up�rqei x ∈ A pou na ikanopoieÐ thn 32 < x < 2.

7. L�joc. P�rte, gia par�deigma, A = (0, 1). Tìte, sup A− inf A = 1− 0 = 1.An ìmwc x, y ∈ (0, 1) tìte 0 < x < 1 kai −1 < −y < 0, �ra −1 < x − y < 1.Dhlad , gia k�je x, y ∈ (0, 1) èqoume x− y 6= 1.

132 · Upodeixeic gia tic Askhseic

8. Swstì. 'Estw A to sÔnolo ìlwn twn r ∈ Q pou ikanopoioÔn thn x < r < y

(gnwrÐzete ìti to A eÐnai mh kenì). Ac upojèsoume ìti to A èqei peperasmènato pl joc stoiqeÐa, ta r1 < · · · < rm. 'Eqoume x < r1, �ra up�rqei rhtìc r∗pou ikanopoieÐ thn x < r∗ < r1. 'Omwc tìte, x < r∗ < y kai r∗ /∈ {r1, . . . , rm}(�topo).


1. (a) Apagwg se �topo. Upojètoume ìti y < x. Tìte, epilègontac ε = x−y2 >

0 èqoumex− (y + ε) = x− y − x− y

2=

x− y

2> 0,

dhlad x > y + ε. 'Atopo.(b) Me ton Ðdio akrib¸c trìpo.(g) JumhjeÐte ìti an a, b ∈ R kai b ≥ 0, tìte |a| ≤ b an kai mìno an −b ≤ a ≤ b.Apì thn upìjesh, gia k�je ε > 0 isqÔoun oi

x ≤ y + ε kai y ≤ x + ε.

Apì to (b) èpetai ìti x ≤ y kai y ≤ x. 'Ara, x = y.(d) AfoÔ a < x < b kai −b < −y < −a, èqoume −(b− a) < x− y < b− a. 'Ara,|x− y| < b− a.

2. (a) P�rte san x to mèso tou diast matoc [a, b]:

x = a +b− a

2=

a + b

2.

(b) Trigwnik anisìthta gia thn apìluth tim .(g) To sÔnolo twn x ∈ R pou ikanopoioÔn thn |a−x| < ε

2 eÐnai to(a− ε

2 , a+ ε2

).

OmoÐwc, to sÔnolo twn x ∈ R pou ikanopoioÔn thn |b−x| < ε2 eÐnai to

(b− ε

2 , b+ ε2

).

Parathr ste ìti b− ε2 < a + ε

2 .

3. Me epagwg . Gia to epagwgikì b ma, upojèste ìti gia k�poion m ∈ N o7m − 4m eÐnai pollapl�sio tou 3 kai gr�yte

7m+1 − 4m+1 = 3 · 7m + 4 · 7m − 4 · 4m.

4. (ii) Merikèc dokimèc ja sac peÐsoun ìti h 2n > n2 isqÔei gia n = 1, denisqÔei gia n = 2, 3, 4 kai (m�llon) isqÔei gia k�je n ≥ 5. DeÐxte me epagwg ìti h 2n > n2 isqÔei gia k�je n ≥ 5: gia to epagwgikì b ma upojètoume ìti h2m > m2 isqÔei gia k�poion m ≥ 5. Tìte,

2m+1 > 2m2 > (m + 1)2

an isqÔei h anisìthta1 + 2m < m2.

6.1 Pragmatikoi Arijmoi · 133

'Omwc, afoÔ m ≥ 5, èqoume

1 + 2m < m + 2m = 3m < m2.

(iv) DeÐxte me epagwg ìti n! > 2n gia k�je n ≥ 4. Elègxte ìti n! ≤ 2n ann = 1, 2, 3.(v) DeÐxte me epagwg ìti 2n−1 > n2 gia k�je n ≥ 7. Elègxte ìti 2n−1 ≤ n2 ann = 1, 2, 3, 4, 5, 6.

5. Me epagwg . Gia to epagwgikì b ma, upojèste ìti

am − bm = (a− b)

(m−1∑

k=0

akbm−1−k

)

gia k�poion m ∈ N. Tìte,

am+1 − bm+1 = (a− b)am + b(am − bm)

= (a− b)am + (a− b)b

(m−1∑

k=0

akbm−1−k

)

= (a− b)

(am +

m−1∑

k=0

akbm−1−kb

)

= (a− b)

(am +

m−1∑

k=0

akbm−k

)

= (a− b)

(m∑

k=0

akbm−k

).

An 0 < a < b, tìte an−1 ≤ akbn−1−k ≤ bn−1 gia k�je k = 0, 1, . . . , n− 1. 'Ara,

nan−1 ≤n−1∑

k=0

akbn−1−k =bn − an

b− a≤ nbn−1.

7. (a) (anisìthta tou Bernoulli). Gia n = 1 h anisìthta isqÔei wc isìthta:1 + a = 1 + a. DeÐqnoume to epagwgikì b ma:

Upojètoume ìti (1+a)m ≥ 1+ma. AfoÔ 1+a ≥ 0, èqoume (1+a)(1+a)m ≥(1 + a)(1 + ma). 'Ara,

(1 + a)m+1 ≥ (1 + a)(1 + ma) = 1 + (m + 1)a + ma2 ≥ 1 + (m + 1)a.

(b) Gia to epagwgikì b ma, parathr ste pr¸ta ìti an 0 < a < 1m+1 tìte èqoume

kai 0 < a < 1m . Apì thn epagwgik upìjesh,

(1+a)m+1(1−(m+1)a) = (1+a)(1−(m+1)a)(1+a)m <(1 + a)(1− (m + 1)a)

1−ma.

'Omwc,

(1+a)(1−(m+1)a) = 1+a−(m+1)a−(m+1)a2 = 1−ma−(m+1)a2 < 1−ma.

'Epetai ìti(1 + a)m+1(1− (m + 1)a) < 1.

8. (a) IsodÔnama, deÐxte ìti an 0 < x < 1, tìte (1 − x)n ≤ 1 − nx + n(n−1)2 x2

gia k�je n ∈ N. Me epagwg . An n = 1 tìte isqÔei san isìthta.Upojèste ìti (1− x)m ≤ 1−mx + m(m−1)

2 x2 gia k�poion m ∈ N. Tìte,

(1− x)m+1 = (1− x)m(1− x)

≤(

1−mx +m(m− 1)

2x2

)(1− x)

= 1− (m + 1)x +[m(m− 1)

2+ m

]x2 − m(m− 1)

2x3

< 1− (m + 1)x +(m + 1)m

2x2,

afoÔ m(m−1)2 + m = (m+1)m

2 kai x > 0.

(b) An n = 1, h anisìthta isqÔei san isìthta. An n ≥ 2, parathr ste ìti apìto diwnumikì an�ptugma,

(1 + a)n =n∑

k=0

(n

k

)ak ≥ 1 + na +

(n

2

)a2 = 1 + na +

n(n− 1)2

a2.

PoÔ qrhsimopoi jhke h upìjesh ìti a > 0?

9. Gia thn pr¸th anisìthta parathroÔme ìti(

1 +1n

)n

<

(1 +

1n + 1

)n+1

⇐⇒(

n + 1n

)n

<

(n + 2n + 1

)nn + 2n + 1

⇐⇒ n + 1n + 2

<

(n(n + 2)(n + 1)2

)n

⇐⇒ 1− 1n + 2

<

(1− 1

(n + 1)2

)n

.

Apì thn anisìthta tou Bernoulli èqoume(

1− 1(n + 1)2

)n

> 1− n

(n + 1)2.

ArkeÐ loipìn na elègxete ìti

n

(n + 1)2<

1n + 2

.

10. (anisìthta arijmhtikoÔ�gewmetrikoÔ mèsou). Pr¸toc trìpoc. DeÐqnoumepr¸ta epagwgik� ìti, gia k�je k ∈ N, an x1, . . . , x2k eÐnai jetikoÐ pragmatikoÐarijmoÐ tìte

2k√x1 · · ·x2k ≤ x1 + · · ·+ x2k

2k.


Gia k = 1 prèpei na elègxoume ìti an x1, x2 > 0 tìte √x1x2 ≤ x1+x22 . Aut

h anisìthta isqÔei an kai mìno an 4x1x2 ≤ (x1 + x2)2, h opoÐa isqÔei diìti(x1 − x2)2 ≥ 0.

Upojètoume ìti an y1, . . . , y2m eÐnai jetikoÐ pragmatikoÐ arijmoÐ tìte

2m√y1 · · · y2m ≤ y1 + · · ·+ y2m

2m.

'Estw x1, . . . , x2m , x2m+1, . . . , x2m+1 > 0. Tìte, efarmìzontac thn epagwgik upìjesh gia touc x1, . . . , x2m > 0 kai x2m+1, . . . , x2m+1 > 0, paÐrnoume

2m+1√x1 · · ·x2mx2m+1 · · ·x2m+1 =√

2m√x1 · · ·x2m · 2m√x2m+1 · · ·x2m+1

≤2m√x1 · · ·x2m + 2m√x2m+1 · · ·x2m+1

2

≤ 12

(x1 + · · ·+ x2m

2m+

x2m+1 + · · ·+ x2m+1

2m

)

=x1 + · · ·+ x2m+1

2m+1.

'Eqoume loipìn deÐxei to zhtoÔmeno an to pl joc N twn arijm¸n eÐnai N = 2k, k ∈N. 'Estw n ∈ N kai èstw x1, . . . , xn > 0. Up�rqei N = 2k > n (exhg ste giatÐ).JewroÔme th N��da x1, . . . , xn, α, . . . , α, ìpou p rame N − n forèc ton jetikìarijmì α = n

√x1 · · ·xn. MporoÔme na efarmìsoume thn anisìthta arijmhtikoÔ�

gewmetrikoÔ mèsou gi' aut th N��da:

N√

x1 · · ·xn · αN−n ≤ x1 + · · ·+ xn + (N − n)αN

.

AfoÔ x1 · · ·xn = αn, h anisìthta paÐrnei th morf

α = N√

αn · αN−n ≤ x1 + · · ·+ xn + (N − n)αN

,

dhlad

Nα ≤ (x1 + · · ·+ xn) + (N − n)α =⇒ nα ≤ x1 + · · ·+ xn.

Sunep¸c,n√

x1 · · ·xn = α ≤ x1 + · · ·+ xn

n.

DeÔteroc trìpoc. Jètoume α = n√

x1x2 · · ·xn kai orÐzoume bk = xk

α , k = 1, . . . , n.ParathroÔme ìti oi bk eÐnai jetikoÐ pragmatikoÐ arijmoÐ me ginìmeno

b1 · · · bn =x1

α· · · xn

α=

x1 · · ·xn

αn= 1.

EpÐshc, h zhtoÔmenh anisìthta paÐrnei th morf

b1 + · · ·+ bn ≥ n.

ArkeÐ loipìn na deÐxoume to ex c:

'Estw n ∈ N. An b1, . . . , bn eÐnai jetikoÐ pragmatikoÐ arijmoÐ meginìmeno b1 · · · bn = 1, tìte b1 + · · ·+ bn ≥ n.

DeÐxte thn me epagwg wc proc to pl joc twn bk: an n = 1 tìte èqoume ènanmìno arijmì, ton b1 = 1. Sunep¸c, h anisìthta eÐnai tetrimmènh: 1 ≥ 1.

Upojètoume ìti gia k�je m��da jetik¸n arijm¸n y1, . . . , ym me ginìmenoy1 · · · ym = 1 isqÔei h anisìthta

y1 + · · ·+ ym ≥ m,

kai deÐqnoume ìti an b1, · · · , bm+1 eÐnai (m + 1) jetikoÐ pragmatikoÐ arijmoÐ meginìmeno b1 · · · bm+1 = 1 tìte

b1 + · · ·+ bm+1 ≥ m + 1.

MporoÔme na upojèsoume ìti b1 ≤ b2 ≤ · · · ≤ bm+1. ParathroÔme ìti, an b1 =b2 = · · · = bm+1 = 1 tìte h anisìthta isqÔei san isìthta. An ìqi, anagkastik�èqoume b1 < 1 < bm+1 (exhg ste giatÐ).

JewroÔme thn m-�da jetik¸n arijm¸n

y1 = b1bm+1 , y2 = b2, . . . , ym = bm.

AfoÔ y1 · · · ym = b1 · · · bm+1 = 1, apì thn epagwgik upìjesh paÐrnoume

(b1bm+1) + b2 + · · ·+ bm = y1 + · · ·+ ym ≥ m.

'Omwc, apì thn b1 < 1 < bm+1 èpetai ìti (bm+1−1)(1−b1) > 0 dhlad b1+bm+1 >

1 + bm+1b1. 'Ara,

b1 + bm+1 + b2 + · · ·+ bm > 1 + b1bm+1 + b2 + · · ·+ bm ≥ 1 + m.

'Eqoume loipìn deÐxei to epagwgikì b ma.An oi x1, · · · , xn den eÐnai ìloi Ðsoi, tìte h apìdeixh pou prohg jhke deÐqnei ìti

h anisìthta eÐnai gn sia (exhg ste giatÐ). Dhlad : sthn anisìthta arijmhtikoÔ�gewmetrikoÔ mèsou isqÔei isìthta an kai mìnon an x1 = · · · = xn.

Gia thn anisìthta

x1x2 · · ·xn ≥(

n1x1

+ · · ·+ 1xn

)n

efarmìste thn anisìthta pou mìlic deÐxame gia touc jetikoÔc pragmatikoÔc a-rijmoÔc 1

x1, . . . , 1

xn.

11. 'Estw A mh kenì k�tw fragmèno uposÔnolo tou R. Jewr ste to sÔnoloB = {−x : x ∈ A}. DeÐxte ìti to B eÐnai mh kenì kai �nw fragmèno. Apì toaxÐwma thc plhrìthtac up�rqei to el�qisto �nw fr�gma s = sup B tou B. DeÐxteìti o −s eÐnai k�tw fr�gma tou A. An y > −s, tìte −y < s. AfoÔ s = sup B,up�rqei b ∈ B tètoio ¸ste −y < b. Tìte, −b ∈ A kai −b < y. Dhlad , o −s

eÐnai k�tw fr�gma tou A kai an y > −s tìte o y den eÐnai k�tw fr�gma tou A.'Epetai ìti −s = inf A.'Alloc trìpoc: OrÐzoume Γ = {x ∈ R : x k�tw fr�gma tou A}. DeÐxte ìti toΓ eÐnai mh kenì kai �nw fragmèno (opoiod pote stoiqeÐo tou A eÐnai èna �nwfr�gma tou Γ). Apì to axÐwma thc plhrìthtac up�rqei to el�qisto �nw fr�gmas = sup Γ tou Γ. DeÐxte ìti o s eÐnai k�tw fr�gma tou A. Tìte, o s eÐnai tomègisto stoiqeÐo tou Γ, dhlad to mègisto k�tw fr�gma tou A.

Gia na deÐxete ìti o s = supΓ eÐnai k�tw fr�gma tou A, prèpei na deÐxeteìti to tuqìn a ∈ A ikanopoieÐ thn supΓ ≤ a. ArkeÐ na deÐxete ìti o a eÐnai �nwfr�gma tou Γ (exhg ste giatÐ). 'Omwc, an x ∈ Γ tìte x ≤ a (apì ton orismì touΓ, o x eÐnai k�tw fr�gma tou A).

12. Apì thn upìjesh, gia k�je a ∈ A èqoume a ≤ a0. 'Ara, to A eÐnai �nwfragmèno kai o a0 eÐnai èna �nw fr�gma tou. 'Epetai ìti to sup A up�rqei, kaisupA ≤ a0.

Apì thn �llh pleur�, afoÔ a0 ∈ A kai o sup A eÐnai �nw fr�gma tou A,èqoume a0 ≤ sup A. Sundu�zontac ta parap�nw blèpoume ìti a0 = sup A.

13. An to A èqei perissìtera apì èna stoiqeÐa, tìte inf A < supB: up�rqounx, y ∈ A me x < y, opìte inf A ≤ x < y ≤ supA. An to A eÐnai monosÔnolo,dhlad A = {a} gia k�poion a ∈ R, tìte inf A = sup A = a (giatÐ?).

'Ara, inf A = sup A an kai mìno an to A eÐnai monosÔnolo.

15. (a) Gr�yte to A sth morf

A ={

2− 12k

: k ∈ N}∪

{1

2k − 1: k ∈ N

}.

Gia k�je a ∈ A isqÔei 0 < a < 2. An y > 0 tìte up�rqei k ∈ N ¸ste 12k−1 < y.

An y < 2 tìte up�rqei k ∈ N ¸ste 2 − 12k > y. Apì ta parap�nw èpetai ìti

inf A = 0 kai sup A = 2 (exhg ste giatÐ). DeÐxte ìti to A den èqei mègisto oÔteel�qisto stoiqeÐo.(b) Gia k�je n,m ∈ N isqÔei 1

2n + 13m ≤ 1

2 + 13 . 'Ara, supB = max B = 5

6 . Giak�je b ∈ B isqÔei b > 0. EpÐshc, an y > 0 tìte up�rqei n ∈ N tètoioc ¸ste

12n

+13n

<22n

=1

2n−1≤ 1

n< y.

'Epetai ìti inf B = 0 (exhg ste giatÐ). DeÐxte ìti to B den èqei el�qisto stoi-qeÐo.

16. Gia k�je m,n ∈ N èqoume∣∣∣ (−1)nm

n+m

∣∣∣ = mn+m < 1. Sunep¸c, A ⊆ (−1, 1).

DeÐxte ìti sup A = 1 kai inf A = −1. Tèloc, deÐxte ìti to A den èqei mègistooÔte el�qisto stoiqeÐo.

17. DeÐqnoume thn inf B ≤ inf A. ArkeÐ na deÐxoume ìti o inf B eÐnai k�twfr�gma tou A. 'Omwc, an x ∈ A tìte x ∈ B (diìti A ⊆ B), �ra inf B ≤ x.

18. (a) Apì thn 'Askhsh 17 èqoume sup(A∪B) ≥ sup A kai sup(A∪B) ≥ sup B.'Ara, sup(A ∪B) ≥ max{supA, sup B}.

Gia thn antÐstrofh anisìthta arkeÐ na deÐxoume ìti o M := max{supA, supB}eÐnai �nw fr�gma tou A ∪ B. 'Estw x ∈ A ∪ B. Tìte, o x an kei se toul�qi-ston èna apì ta A B. An x ∈ A tìte x ≤ supA ≤ M kai an x ∈ B tìtex ≤ supB ≤ M .

(b) IsqÔei h anisìthta sup(A ∩ B) ≤ min{sup A, sup B}. MporeÐ ìmwc na eÐnaign sia (d¸ste par�deigma).

19. Upojètoume pr¸ta ìti sup A ≤ inf B. Tìte, gia k�je a ∈ A kai gia k�jeb ∈ B isqÔei a ≤ supA ≤ inf B ≤ b.

AntÐstrofa, upojètoume ìti gia k�je a ∈ A kai gia k�je b ∈ B isqÔei a ≤ b.Ja deÐxoume ìti supA ≤ inf B.

Pr¸toc trìpoc: Gia na deÐxoume ìti sup A ≤ inf B, arkeÐ na deÐxoume ìti o inf B

eÐnai �nw fr�gma tou A. 'Estw a ∈ A. Apì thn upìjesh, gia k�je b ∈ B isqÔeia ≤ b. 'Ara, o a eÐnai k�tw fr�gma tou B. Sunep¸c, a ≤ inf B. To a ∈ A tantuqìn, �ra o inf B eÐnai �nw fr�gma tou A.

DeÔteroc trìpoc: Ac upojèsoume ìti inf B 0 me thnidiìthta inf B + ε < supA − ε (exhg ste giatÐ). Apì ton qarakthrismì touinfimum, up�rqei b ∈ B pou ikanopoieÐ thn b < inf B + ε kai up�rqei a ∈ A pouikanopoieÐ thn supA− ε < a. Tìte, b < inf B + ε < supA− ε < a. Autì èrqetaise antÐfash me thn upìjesh.

20. Apì thn upìjesh èpetai ìti A ∩ B = ∅ (exhg ste giatÐ). EpÐshc, apì thn'Askhsh 19 èqoume γ := sup A ≤ δ := inf B. Dikaiolog ste diadoqik� ta ex c:

(i) γ = δ: an eÐqame γ < δ tìte o γ+δ2 den ja an ke sto A ∪ B (exhg ste

giatÐ).

(ii) (−∞, γ] ⊇ A kai [γ,∞) ⊇ B.

(iii) (−∞, γ) ⊆ A kai (γ,∞) ⊆ B.

(iv) O γ an kei se akrib¸c èna apì ta A B.

21. Pr¸toc trìpoc: Gia na deÐxoume ìti supA ≤ supB, arkeÐ na deÐxoume ìti osup B eÐnai �nw fr�gma tou A. 'Estw a ∈ A. Apì thn upìjesh, up�rqei b ∈ B

¸stea ≤ b ≤ sup B.

To a ∈ A tan tuqìn, �ra o sup B eÐnai �nw fr�gma tou A.

DeÔteroc trìpoc: Ac upojèsoume ìti supA > sup B. Up�rqei ε > 0 me thnidiìthta sup A − ε > sup B (giatÐ?). Apì ton qarakthrismì tou supremum,up�rqei a ∈ A pou ikanopoieÐ thn a > sup A− ε. Tìte, gia k�je b ∈ B èqoume

b ≤ supB < supA− ε < a.

Autì èrqetai se antÐfash me thn upìjesh.

22. Jètoume A = (a, b) ∩ Q = {x ∈ Q : a < x < b}. Apì ton orismì tou A

èqoume x a.Upojètoume ìti sup A < b. Apì thn puknìthta tou Q sto R up�rqei q ∈ Q ¸stesupA < q < b. Tìte a < q < b, dhlad q ∈ A. Autì eÐnai �topo, lìgw thcsupA < q. 'Ara, sup A = b.

Me an�logo epiqeÐrhma deÐxte ìti inf A = a.

23. Ac upojèsoume ìti x 6= y. QwrÐc periorismì thc genikìthtac, mporoÔme naupojèsoume ìti x < y. Up�rqei q ∈ Q me thn idiìthta x < q < y. Tìte, q ∈ Ay

kai q /∈ Ax. 'Ara, Ax 6= Ay.An x = y, eÐnai fanerì ìti: gia k�je q ∈ Q isqÔei q < x ⇐⇒ q < y. 'Ara,Ax = Ay.

G. Ask seic

1. (a) (anisìthta Cauchy–Schwarz). H pio fusiologik apìdeixh eÐnai me epa-gwg : parathr ste pr¸ta ìti arkeÐ na deÐxoume thn anisìthta sthn perÐptwshpou ak ≥ 0, bk ≥ 0 gia k�je k = 1, . . . , n (exhg ste giatÐ).n = 2: Elègxte ìti gia k�je a1, a2, b1, b2 ∈ R isqÔei h anisìthta

(a1b1 + a2b2)2 ≤ (a21 + a2

2)(b21 + b2

2).

Epagwgikì b ma. Upojètoume ìti to zhtoÔmeno isqÔei gia opoiesd pote dÔo k��dec pragmatik¸n arijm¸n, k = 2, . . . ,m. 'Estw a1, . . . , am+1 kai b1, . . . , bm+1

mh arnhtikoÐ pragmatikoÐ arijmoÐ. Qrhsimopoi¸ntac thn epagwgik upìjesh gr�-foume

m+1∑

k=1

akbk =m∑

k=1

akbk + am+1bm+1

≤(

m∑

k=1

a2k

)1/2 (m∑

k=1

b2k

)1/2

+ am+1bm+1.

An orÐsoume x =(∑m

k=1 a2k

)1/2 kai y =(∑m

k=1 b2k

)1/2, tìte (apì to b ma n = 2)èqoume(

m∑

k=1

a2k

)1/2 (m∑

k=1

b2k

)1/2

+ am+1bm+1 = xy + am+1bm+1

≤ (x2 + a2m+1)

1/2(y2 + b2m+1)

1/2

=(a21 + · · ·+ a2

m + a2m+1

)1/2 (b21 + · · ·+ b2

m + b2m+1

)1/2.

Sundu�zontac ta parap�nw blèpoume ìti

m+1∑

k=1

akbk ≤(a21 + · · ·+ a2

m + a2m+1

)1/2 (b21 + · · ·+ b2

m + b2m+1

)1/2.


'Etsi, èqoume apodeÐxei to epagwgikì b ma.

(b) (anisìthta tou Minkowski). Qrhsimopoi¸ntac thn anisìthta Cauchy–Schwarz,gr�foume

n∑

k=1

(ak + bk)2 =n∑

k=1

ak(ak + bk) +n∑

k=1

bk(ak + bk)

≤(

n∑

k=1

a2k

)1/2 (n∑

k=1

(ak + bk)2)1/2

+

(n∑

k=1

b2k

)1/2 (n∑

k=1

(ak + bk)2)1/2

=

(n∑

k=1

a2k

)1/2

+

(n∑

k=1

b2k

)1/2

(n∑

k=1

(ak + bk)2)1/2

.

'Epetai to zhtoÔmeno (exhg ste giatÐ).

2. (tautìthta tou Lagrange). Parathr ste ìti

(n∑

k=1

a2k

)(n∑

k=1

b2k

)=

(n∑

k=1

a2k

)

n∑

j=1

b2j

=

n∑

k,j=1

a2kb2

j

kai, ìmoia,

(n∑

k=1

a2k

)(n∑

k=1

b2k

)=

n∑

j=1

a2j

(n∑

k=1

b2k

)=

n∑

k,j=1

a2jb

2k.

EpÐshc,

(n∑

k=1

akbk

)2

=

(n∑

k=1

akbk

)

n∑

j=1

ajbj

=

n∑

k,j=1

akbjajbk.

'Ara, to aristerì mèloc isoÔtai (exhg ste giatÐ) me

12

n∑

k,j=1

(a2kb2

j − 2akbjajbk + a2jb

2k) =

12

n∑

k,j=1

(akbj − ajbk)2.

H anisìthta Cauchy–Schwarz prokÔptei �mesa.

3. Apì thn anisìthta arijmhtikoÔ�gewmetrikoÔ�armonikoÔ mèsou èqoume

a1 + a2 + · · ·+ an

n≥ n√

a1a2 · · · an ≥ n1a1

+ 1a2

+ · · ·+ 1an

.

4. Parathr ste pr¸ta ìti 1 = (a + b)2 ≥ 4ab. Efarmìzontac thn anisìthta


Cauchy-Schwarz paÐrnoume

2

[(a +

1a

)2

+(

b +1b

)2]

= (12 + 12)

[(a +

1a

)2

+(

b +1b

)2]

≥(

1 · (a +1a) + 1 · (b +

1b))2

≥ ((a + 4b) + (b + 4a)

)2

= 25(a + b)2 = 25.

5. Epagwg .

6*. MporeÐte na apodeÐxete tic dÔo anisìthtec me epagwg . Mia polÔ piìsÔntomh apìdeixh eÐnai h ex c.Dexi� anisìthta: Apì thn upìjesh ìti a1 ≥ a2 ≥ · · · ≥ an kai b1 ≥ b2 ≥ · · · ≥ bn

èpetai (giatÐ?) ìtin∑

k,j=1

(ak − aj)(bk − bj) ≥ 0.

Dhlad ,

(∗)n∑

k,j=1

akbk +n∑

k,j=1

ajbj ≥n∑

k,j=1

ajbk +n∑

k,j=1

akbj .

Parathr ste ìti

n∑

k,j=1

akbk =n∑

j=1

(n∑

k=1

akbk

)= n(a1b1 + · · ·+ anbn).

'Omoia,n∑

k,j=1

ajbj = n(a1b1 + · · ·+ anbn).

Apì thn �llh pleur�,

n∑

k,j=1

ajbk =n∑

k,j=1

akbj = (a1 + · · ·+ an)(b1 + · · ·+ bn).

'Ara, h (∗) paÐrnei th morf

2n(a1b1 + · · ·+ anbn) ≥ 2(a1 + · · ·+ an)(b1 + · · ·+ bn),

pou eÐnai h zhtoÔmenh anisìthta.Arister anisìthta: Qrhsimopoi ste to gegonìc ìti

n∑

k,j=1

(ak − aj)(bn−k+1 − bn−j+1) ≤ 0.

7*. Jewr ste touc arijmoÔc

bm =m∑

k=1

ak −n∑

k=m+1

ak, m = 1, . . . , n− 1

kai

b0 = −n∑

k=1

ak , bn =n∑

k=1

ak.

An B = {k : 1 ≤ k ≤ n kai bk > 0}, tìte to B eÐnai mh kenì (exhg ste giatÐ).'Ara, èqei el�qisto stoiqeÐo: ac to poÔme m0. Parathr ste ìti m0 ≥ 1. AfoÔo m0 eÐnai to el�qisto stoiqeÐo tou B, èqoume bm0 > 0 kai bm0−1 ≤ 0. 'Epetai(exhg ste giatÐ) ìti

|bm0 |+ |bm0−1| = bm0 − bm0−1 = 2am0 ≤ 2max{a1, . . . , an}.

Autì dÐnei to zhtoÔmeno (exhg ste giatÐ).

8. Ja deÐxoume ìti inf(A + B) = inf A + inf B.

(a) inf(A + B) ≥ inf A + inf B: arkeÐ na deÐxoume ìti o inf A + inf B eÐnai k�twfr�gma tou A+B. 'Estw x ∈ A+B. Up�rqoun a ∈ A kai b ∈ B ¸ste x = a+b.'Omwc, a ≥ inf A kai b ≥ inf B. 'Ara,

x = a + b ≥ inf A + inf B.

To x ∈ A + B tan tuqìn, �ra o inf A + inf B eÐnai k�tw fr�gma tou A + B.

(b) inf(A + B) ≤ inf A + inf B: arkeÐ na deÐxoume ìti o inf(A + B)− inf B eÐnaik�tw fr�gma tou A. Gia to skopì autì, jewroÔme (tuqìn) a ∈ A kai deÐqnoumeìti

inf(A + B)− a ≤ inf B.

Gia thn teleutaÐa anisìthta arkeÐ na deÐxoume ìti o inf(A + B) − a eÐnai k�twfr�gma tou B: èstw b ∈ B. Tìte, a + b ∈ A + B. 'Ara,

inf(A + B) ≤ a + b =⇒ inf(A + B)− a ≤ b.

To b ∈ B tan tuqìn, �ra o inf(A + B)− a eÐnai k�tw fr�gma tou B.

DeÔteroc trìpoc: 'Estw ε > 0. Apì ton qarakthrismì tou supremum kai touinfimum, up�rqoun a ∈ A kai b ∈ B pou ikanopoioÔn tic

a < inf A + ε kai b < inf B + ε.

Tìte, afoÔ a + b ∈ A + B,

inf(A + B) ≤ a + b < inf A + inf B + 2ε.

To ε > 0 tan tuqìn, �ra inf(A + B) ≤ inf A + inf B.

6.2 Akoloujiec pragmatikwn arijmwn · 143




1. L�joc. H akoloujÐa an = (−1)n eÐnai fragmènh all� den sugklÐnei.

2. Swstì. Upojètoume ìti an → a ∈ R. PaÐrnoume ε = 1 > 0. MporoÔme nabroÔme n0 ∈ N ¸ste |an − a| < 1 gia k�je n ≥ n0. Dhlad ,

an n ≥ n0, tìte |an| < 1 + |a|.

JètoumeM = max{|a1|, . . . , |an0 |, 1 + |a|}

kai elègqoume ìti |an| ≤ M gia k�je n ∈ N (diakrÐnete peript¸seic: n ≤ n0 kain > n0). 'Ara, h (an) eÐnai fragmènh.

3. Swstì. Upojètoume ìti h (an) sugklÐnei ston pragmatikì arijmì a. Epilè-goume ε = 1

2 kai brÐskoume n0 ∈ N ¸ste: gia k�je n ≥ n0 isqÔei |an − a| < 12 .

Tìte, an n ≥ n0 èqoume

|an − an0 | = |(an − a) + (a− an0)| ≤ |an − a|+ |a− an0 | <12

+12

= 1.

'Omwc, oi an kai an0 eÐnai akèraioi, �ra an = an0 . Dhlad , h (an) eÐnai telik�stajer : gia k�je n ≥ n0 èqoume an = an0 .

H antÐstrofh kateÔjunsh eÐnai apl : an gia mia akoloujÐa (an) sto R up�r-qoun n0 ∈ N kai a ∈ R ¸ste gia k�je n ≥ n0 na isqÔei an = a, tìte an → a

(exhg ste, me b�sh ton orismì tou orÐou).

4. L�joc. Ac upojèsoume ìti (an) eÐnai mia gnhsÐwc fjÐnousa akoloujÐa fusik¸narijm¸n. To sÔnolo A = {an : n ∈ N} eÐnai mh kenì uposÔnolo tou N. Apìthn arq thc kal c di�taxhc, èqei el�qisto stoiqeÐo. Dhlad , up�rqei m ∈ N¸ste: gia k�je n ∈ N isqÔei am ≤ an. Autì eÐnai �topo, afoÔ am+1 ∈ A kaiam+1 < am.

5. L�joc. H akoloujÐa an =√

2n eÐnai akoloujÐa �rrhtwn arijm¸n, ìmwc an → 0.

6. Swstì. DiakrÐnete peript¸seic. An o x eÐnai �rrhtoc, mporeÐte na jewr seteth stajer akoloujÐa an = x. An o x eÐnai rhtìc, mporeÐte na jewr sete thnakoloujÐa an = x +

√2

n .

7. Swstì. Upojètoume pr¸ta ìti an → 0. 'Estw M > 0. AfoÔ an → 0,efarmìzontac ton orismì me ε = 1

M > 0, mporoÔme na broÔme n0 = n0(ε) =n0(M) ∈ N ¸ste: gia k�je n ≥ n0 isqÔei 0 < an < 1

M . Dhlad , up�rqei

n0 = n0(M) ∈ N ¸ste: gia k�je n ≥ n0 isqÔei 1an

> M . 'Epetai ìti 1an→ +∞.

Gia thn antÐstrofh kateÔjunsh ergazìmaste me an�logo trìpo.

8. L�joc. H akoloujÐa an = (−1)n

n sugklÐnei sto 0 all� den eÐnai monìtonh.

9. Swstì. 'Estw M > 0. AfoÔ h (an) den eÐnai �nw fragmènh, up�rqei n0 ∈ N¸ste an0 > M . AfoÔ h (an) eÐnai aÔxousa, gia k�je n ≥ n0 isqÔei an ≥ an0 >

M . AfoÔ o M > 0 tan tuq¸n, sumperaÐnoume ìti an → +∞.

10. L�joc. H an = (−1)n eÐnai fragmènh kai h bn = 1 sugklÐnei, ìmwc hanbn = (−1)n den sugklÐnei.

11. L�joc. H an = (−1)n den sugklÐnei, ìmwc h |an| = 1 sugklÐnei.

12. Swstì. H basik idèa eÐnai ìti, afoÔ a > 0 kai an → a, ja up�rqei n0 methn idiìthta: gia k�je n ≥ n0 isqÔei an > a/2. Dhlad , telik� ìloi oi ìroi thc(an) xepernoÔn ton jetikì arijmì a/2.

Pr�gmati, an efarmìsete ton orismì tou orÐou gia thn (an) me ε = a/2 > 0,mporeÐte na breÐte n0 ∈ R ¸ste: gia k�je n ≥ n0,

|an − a| < ε = a/2 =⇒ a/2 < an < 3a/2.

Tìte, o jetikìc arijmìc m := min{a1, . . . , an0 , a/2} eÐnai k�tw fr�gma tou su-nìlou A = {an : n ∈ N} (exhg ste giatÐ). Sunep¸c, inf(A) ≥ m > 0.

13. Swstì. H basik idèa eÐnai ìti, afoÔ a1 > 0 kai an → 0, ja up�rqei n0 methn idiìthta: gia k�je n ≥ n0 isqÔei an < a1. Dhlad , up�rqei stoiqeÐo tou A

megalÔtero apì {ìla} (ektìc apì peperasmèna to pl joc) ta stoiqeÐa tou A.Pr�gmati, an efarmìsete ton orismì tou orÐou gia thn (an) me ε = a1 > 0,

mporeÐte na breÐte n0 ∈ R ¸ste: gia k�je n ≥ n0,

an = |an − 0| < ε = a1.

Tìte, o megalÔteroc apì touc a1, . . . , an0 eÐnai to mègisto stoiqeÐo tou A: an keisto A kai eÐnai megalÔteroc Ðsoc apì k�je an (exhg ste giatÐ).

14. L�joc. Jewr ste thn akoloujÐa (an) me a2k = k kai a2k−1 = 1 gia k�jek ∈ N. Tìte, an > 0 kai h (an) den eÐnai �nw fragmènh, ìmwc an → +∞ (anautì Ðsque, ja èprepe ìloi telik� oi ìroi thc (an) na eÐnai megalÔteroi apì 2, toopoÐo den isqÔei afoÔ ìloi oi perittoÐ ìroi thc eÐnai Ðsoi me 1).

15. L�joc. Qrhsimopoi ste to par�deigma thc prohgoÔmenhc er¸thshc gia nadeÐxete ìti mporeÐ na isqÔei h prìtash {gia k�je M > 0 up�rqoun �peiroi ìroithc (an) pou eÐnai megalÔteroi apì M} qwrÐc na isqÔei h prìtash {an → +∞}.

H �llh kateÔjunsh eÐnai swst : an an → +∞ tìte (apì ton orismì) giak�je M > 0 ìloi telik� oi ìroi thc (an) eÐnai megalÔteroi apì M . 'Ara, giak�je M > 0 up�rqoun �peiroi ìroi thc (an) pou eÐnai megalÔteroi apì M .

16. Swstì. Upojètoume pr¸ta ìti up�rqei upakoloujÐa (akn) thc (an) ¸ste

akn→ +∞. 'Estw M > 0. AfoÔ akn

→ +∞, ìloi telik� oi ìroi akneÐnai

megalÔteroi apì M . Opìte, up�rqoun �peiroi ìroi thc (an) pou eÐnai megalÔteroiapì M . AfoÔ o M > 0 tan tuq¸n, h (an) den eÐnai �nw fragmènh.

'Alloc trìpoc: an h (an) tan �nw fragmènh, tìte kai k�je upakoloujÐathc (an) ja tan �nw fragmènh. Tìte ìmwc, h (an) den ja mporoÔse na èqeiupakoloujÐa h opoÐa na teÐnei sto +∞.

AntÐstrofa, ac upojèsoume ìti h (an) den eÐnai �nw fragmènh. Ja broÔme e-pagwgik� k1 < · · · < kn < kn+1 < · · · ¸ste akn

> n. Tìte, gia thn upakoloujÐa(akn

) thc (an) ja èqoume akn→ +∞:

AfoÔ h (an) den eÐnai �nw fragmènh, mporoÔme na broÔme k1 ∈ N ¸steak1 > 1. Ac upojèsoume ìti èqoume breÐ k1 < · · · < km ¸ste akj > j gia k�jej = 1, . . . , m. Jètoume M = max{a1, a2, . . . , akm

, m + 1}. AfoÔ h (an) deneÐnai �nw fragmènh, mporoÔme na broÔme km+1 ∈ N ¸ste akm+1 > M . Apì tonorismì tou M èqoume akm+1 > m + 1 kai akm+1 > an gia k�je n = 1, 2, . . . , km.Sunep¸c, km+1 > km. Autì oloklhr¸nei to epagwgikì b ma.

17. Swstì. Upojètoume ìti an → a. 'Estw (bn) = (akn) mia upakoloujÐa thc(an) kai èstw ε > 0. Up�rqei n0 ∈ N ¸ste: gia k�je n ≥ n0 isqÔei |an−a| < ε.Parathr ste ìti kn0 ≥ n0 (diìti h (kn) eÐnai gnhsÐwc aÔxousa akoloujÐa fusik¸narijm¸n). Tìte, gia k�je n ≥ n0 èqoume kn ≥ n0, �ra |bn − a| = |akn − a| < ε.'Epetai ìti bn → a.

18. Swstì. JumhjeÐte ton orismì tou shmeÐou koruf c miac akoloujÐac (an):h (an) èqei shmeÐo koruf c ton ìro ak an ak ≥ am gia k�je m ≥ k.

AfoÔ h (an) den èqei fjÐnousa upakoloujÐa, den mporeÐ na èqei �peira shmeÐakoruf c: pr�gmati, ac upojèsoume ìti up�rqoun k1 < k2 < · · · < kn < · · · ¸steoi ìroi ak1 , . . . , akn , . . . na eÐnai shmeÐa koruf c thc (an). Tìte,

ak1 ≥ ak2 ≥ · · · ≥ akn ≥ · · · ,

dhlad h upakoloujÐa (akn) eÐnai fjÐnousa.

'Ara, h (an) èqei peperasmèna to pl joc shmeÐa koruf c: Up�rqei dhlad k1 ∈ N (to teleutaÐo shmeÐo koruf c o k1 = 1 an den up�rqoun shmeÐa koruf c)me thn idiìthta: an n ≥ k1, up�rqei n′ > n ¸ste an′ > an.

BrÐskoume k2 > k1 ¸ste ak1 < ak2 , katìpin brÐskoume k3 > k2 ¸ste ak2 >

ak3 kai oÔtw kajex c. Up�rqoun dhlad k1 < k2 < · · · < kn < · · · ¸ste

ak1 < ak2 < · · · < akn < · · · .

'Ara, h (an) èqei toul�qiston mÐa gnhsÐwc aÔxousa upakoloujÐa.

19. L�joc. Jewr ste thn akoloujÐa (an) me a2k = k kai a2k−1 = 1 gia k�jek ∈ N. Tìte, an 6→ 1, ìmwc h (an) den èqei upakoloujÐa pou na sugklÐnei sek�poion pragmatikì arijmì b 6= 1 (exhg ste giatÐ).

ShmeÐwsh. An k�noume thn epiplèon upìjesh ìti h (an) eÐnai fragmènh, tìte hprìtash eÐnai swst . Sthn 'Askhsh 23 parak�tw apodeiknÔetai ìti an h akolou-jÐa (an) den sugklÐnei ston a tìte up�rqoun ε > 0 kai upakoloujÐa (akn

) thc(an) ¸ste: gia k�je n ∈ N isqÔei |akn

− a| ≥ ε. AfoÔ h (an) eÐnai fragmènh, h(akn

) eÐnai epÐshc fragmènh. Apì to je¸rhma Bolzano–Weierstrass h (akn) èqei

upakoloujÐa (akln) h opoÐa sugklÐnei se k�poion b ∈ R. 'Omwc, |akln

− a| ≥ ε

gia k�je n ∈ N, �ra |b− a| = limn→∞ |akln− a| ≥ ε. Dhlad , b 6= a.

20. L�joc. Apì to je¸rhma Bolzano–Weierstrass, k�je fragmènh akoloujÐaèqei sugklÐnousa upakoloujÐa.

21. L�joc. Jewr ste thn akoloujÐa (an) me a2k = k kai a2k−1 = 1 gia k�jek ∈ N. H (an) den eÐnai fragmènh, ìmwc h upakoloujÐa (a2k−1) eÐnai stajer (�ra, fragmènh).

22. Swstì. Jewr ste mia upakoloujÐa (akn) thc (an). An n ∈ N tìte kn <

kn+1. H (an) eÐnai aÔxousa, �ra: an s, t ∈ N kai s < t tìte as ≤ at (exhg stegiatÐ). PaÐrnontac s = kn kai t = kn+1 sumperaÐnoume ìti akn ≤ akn+1 .

23. Swstì. Upojètoume ìti h (an) eÐnai aÔxousa kai ìti up�rqei upakoloujÐa(akn) thc (an) h opoÐa sugklÐnei ston a ∈ R.

AfoÔ akn → a, h (akn) eÐnai fragmènh. Sunep¸c, up�rqei M ∈ R tètoioc¸ste: gia k�je n ∈ N isqÔei akn ≤ M . Ja deÐxoume ìti: gia k�je n ∈ N isqÔeian ≤ M . Tìte, h (an) eÐnai aÔxousa kai �nw fragmènh, �ra sugklÐnei.

Jewr ste tuqìnta n ∈ N. AfoÔ n ≤ kn kai h (an) eÐnai aÔxousa, èqoumean ≤ akn ≤ M .

24. Swstì. Ja broÔme epagwgik� k1 < · · · < kn < kn+1 < · · · ¸ste |akn | < 1n3 .

Tìte, gia thn upakoloujÐa (akn) thc (an) ja èqoume n2akn → 0 (exhg ste giatÐ).AfoÔ h an → 0, mporoÔme na broÔme k1 ∈ N ¸ste |ak1 | < 1. Ac upojèsoume

ìti èqoume breÐ k1 < · · · < km ¸ste |akj | < 1j3 gia k�je j = 1, . . . , m. Jètoume

ε = 1(m+1)3 > 0. AfoÔ h an → 0, mporoÔme na broÔme n0 ∈ N ¸ste |an| < 1

(m+1)3

gia k�je n ≥ n0. Eidikìtera, up�rqei km+1 > km ¸ste |akm+1 | < 1(m+1)3 . Autì

oloklhr¸nei to epagwgikì b ma.

25. Swstì. Jètoume A = {an : n ∈ N}. Apì ton basikì qarakthrismì tousupremum, gia k�je ε > 0 up�rqei x = x(ε) ∈ A ¸ste 1 − ε < x ≤ 1. AfoÔ1 /∈ A, èqoume thn isqurìterh anisìthta 1− ε < x < 1.

Qrhsimopoi¸ntac to parap�nw, ja broÔme epagwgik� k1 < · · · < kn < kn+1 <

· · · ¸ste ak1 < · · · < akn < akn+1 < · · · kai 1 − 1n < akn < 1. Tìte, gia thn

gnhsÐwc aÔxousa upakoloujÐa (akn) thc (an) ja èqoume akn → 1.Efarmìzontac ton qarakthrismì tou supremum me ε = 1, brÐskoume ak1 ∈ A

pou ikanopoieÐ thn 0 < ak1 < 1.Ac upojèsoume ìti èqoume breÐ k1 < · · · < km ¸ste ak1 < · · · < akm kai 1−

1j < akj < 1 gia k�je j = 1, . . . , m. Jètoume s = max{1− 1

m+1 , a1, a2, . . . , akm}.AfoÔ s < 1, mporoÔme loipìn na broÔme akm+1 ∈ A pou ikanopoieÐ thn s <

akm+1 < 1. Tìte, km+1 > km, akm< akm+1 kai 1 − 1

m+1 < akm+1 < 1. Autìoloklhr¸nei to epagwgikì b ma.

26. Swstì. An an → a tìte oi upakoloujÐec (a2k) kai (a2k−1) sugklÐnoun stona. Apì thn upìjesh ìti an+2 = an gia k�je n ∈ N, blèpoume ìti oi (a2k) kai(a2k−1) eÐnai stajerèc akoloujÐec: up�rqoun x, y ∈ R ¸ste

x = a1 = a3 = a5 = · · · kai y = a2 = a4 = a6 = · · · .

Apì tic a2k−1 → x kai a2k → y èpetai ìti x = y = a. 'Ara, h (an) eÐnai stajer .


1. (a) AfoÔ n + (−1)n ≥ n − 1, h anisìthta n + (−1)n > 103 isqÔei ann ≥ n0 := 103 + 2. Dhlad , isqÔei telik�.

(b) An o n eÐnai �rtioc kai megalÔteroc apì 1000, tìte (−1)nn = n > 1000. 'Ara,h (−1)nn > 1000 isqÔei gia �peirouc to pl joc fusikoÔc n. An o n eÐnai perittìc,tìte h (−1)nn > 1000 den isqÔei (exhg ste giatÐ). 'Ara, h (−1)nn > 1000 denisqÔei telik�.

(g) 'Eqoume 1n + (−1)n

n2 ≤ 1n + 1

n2 ≤ 2n . An loipìn n > 2 · 1010, tìte 1

n + (−1)n

n2 <

10−10 (dhlad , h anisìthta isqÔei telik�).

(d) H (1+(−1)n)n < 2 isqÔei an kai mìno an o n eÐnai perittìc (exhg ste giatÐ).Sunep¸c, isqÔei gia �peirouc to pl joc fusikoÔc n, ìqi ìmwc telik�.

(e) An n2 + 2n < 106 tìte n2 < 106, dhlad n < 1000. 'Ara, h anisìthta denisqÔei gia �peirouc to pl joc fusikoÔc n (oÔte telik�).

2.(a) PaÐrnontac ε = 0.001 > 0 brÐskoume n1 ∈ N ¸ste: gia k�je n ≥ n1

isqÔei 1.999 < an < 2.001. 'Ara, k�je n ≥ n1 an kei sto A1. To N \ A1 eÐnaipeperasmèno (kai to A1 �peiro, kai m�lista, telikì tm ma tou N).(g) PaÐrnontac ε = 0.02 > 0 brÐskoume n3 ∈ N ¸ste: gia k�je n ≥ n3 isqÔei1.98 < an < 2.02. 'Ara, k�je n ≥ n3 an kei sto N\A3. To A3 eÐnai peperasmèno(kai to N \A3 �peiro, kai m�lista, telikì tm ma tou N).(e) PaÐrnontac ε = 0.00003 > 0 brÐskoume n5 ∈ N ¸ste: gia k�je n ≥ n5 isqÔei1.99997 < an < 2.00003 < 2.0001. 'Ara, k�je n ≥ n5 an kei sto A5. To N \A5

eÐnai peperasmèno (kai to A5 �peiro, kai m�lista, telikì tm ma tou N).(st) To A6 mporeÐ na eÐnai peperasmèno �peiro, exart�tai apì thn (an). P�rtesan paradeÐgmata tic akoloujÐec

an = 2, an = 2 +1n

, an = 2 +(−1)n

n.

'Olec ikanopoioÔn thn limn→∞

an = 2. Sthn pr¸th perÐptwsh èqoume A6 = N kaiN \A6 = ∅. Sthn deÔterh, A6 = ∅ kai N \A6 = N. Sthn trÐth, tìso to A6 ìsokai to N \ A6 eÐnai �peira sÔnola (to sÔnolo twn peritt¸n kai to sÔnolo twn�rtiwn fusik¸n, antÐstoiqa).

3. Ac doÔme gia par�deigma thn (cn): apì tic 2n = (1 + 1)n ≥ 1 + n > n kain2 + 1 ≥ n + 1 > n èqoume 0 < cn < 1

n gia k�je n ∈ N. Pr�gmati, an on eÐnai thc morf c n = 3k + 1 gia k�poion mh arnhtikì akèraio k (dhlad , ann = 1, 4, 7, 10, 13, . . .) tìte 0 < cn = 1

2n < 1n . An p�li n 6= 3k + 1 gia k�je mh

arnhtikì akèraio k, tìte 0 < cn = 1n2+1 < 1

n .An t¸ra mac d¸soun (tuqìn) ε > 0, up�rqei n0 = n0(ε) ∈ N ¸ste 1

n0< ε.

Tìte, gia k�je n ≥ n0 èqoume

−ε < 0 < cn <1n≤ 1

n0< ε,

dhlad , |cn| < ε. Sunep¸c, cn → 0.

4. Parathr ste ìti

|an − 1| =∣∣∣∣n2 − n

n2 + n− 1

∣∣∣∣ =∣∣∣∣−2n

n2 + n

∣∣∣∣ =2n

n2 + n<

2n

n2=

2n

.

Gia tuqìn ε > 0 breÐte n0(ε) ∈ N ¸ste 2n0

< ε. Tìte, gia k�je n ≥ n0 isqÔei|an − 1| < 2

n ≤ 2n0

< ε.

5. Epilègoume ε = a/2 > 0. AfoÔ limn→∞

an = a, up�rqei n0(ε) ∈ N me thnidiìthta: gia k�je n ≥ n0 isqÔei |an − a| < a/2. IsodÔnama, gia k�je n ≥ n0

isqÔei a/2 < an < 3a/2. 'Ara, an > a/2 > 0 telik� (apì ton n0-ostì ìro kaipèra).

6. (a) Ac upojèsoume ìti b < a. Tìte, an − bn → a− b > 0. Apì thn 'Askhsh5, up�rqei n0 ∈ N me thn idiìthta: gia k�je n ≥ n0 isqÔei

an − bn >a− b

2> 0.

'Atopo, afoÔ an ≤ bn gia k�je n ∈ N (apì thn upìjesh).

(b) 'Oqi: an orÐsoume an = 0 kai bn = 1n , tìte an < bn gia k�je n ∈ N, all�

limn→∞

an = limn→∞

bn = 0. Ja èqoume ìmwc a ≤ b (apì to pr¸to er¸thma).

(g) Jewr ste tic stajerèc akoloujÐec γn = m, δn = M kai efarmìste to (a).

8. (g) Jètoume ε = `−12 > 0. AfoÔ an+1

an→ `, up�rqei n0 ∈ N ¸ste: gia k�je

n ≥ n0,an+1

an> `− ε =

` + 12

.




θn0· θn.

AfoÔ limn→∞

(d) Jètoume ε = 1−`2 > 0. AfoÔ

∣∣∣an+1an


an+1

an

∣∣∣∣ < ` + ε =` + 1

2.



|an| < ρn−n0 |an0 | =|an0 |ρn0

· ρn.

AfoÔ limn→∞

ρn = 0, èpetai ìti an → 0.

10. (a) H (|bn|) eÐnai fjÐnousa kai k�tw fragmènh apì to 0. 'Ara, up�rqei x ≥ 0¸ste |bn| → x. Apì thn |bn+1| = |bn| · |a|, paÐrnontac ìrio wc proc n kai stadÔo mèlh, èqoume x = x · |a|. AfoÔ |a| 6= 1, sumperaÐnoume ìti x = 0.(b) Parathr ste ìti

∣∣∣∣1− x2

1 + x2

∣∣∣∣ =|1− x2|1 + x2

≤ 1 + x2

1 + x2= 1.

EpÐshc, an x 6= 0 tìte 1−x2

1+x2 6= ±1 (exhg ste giatÐ), �ra∣∣∣∣1− x2

1 + x2

∣∣∣∣ < 1.

Apì to (a), an x 6= 0 tìte h akoloujÐa(

1−x2

1+x2

)n

→ 0. Tèloc, an x = 0 èqoume(1−x2

1+x2

)n

= 1 → 1.

Dhlad , h akoloujÐa(

1−x2

1+x2

)n

sugklÐnei, ìpoio ki an eÐnai to x.

11. (a) an = n3+5n2+22n3+9 . MporeÐte na gr�yete

an =1 + 5

n + 2n3

2 + 9n3

→ 12.

(b) bn = 1n2 + 1

(n+1)2 + · · · + 1(2n)2 . Parathr ste ìti 1

(2n)2 ≤ 1(n+k)

2 ≤ 1n2 gia

k�je k = 0, . . . , n. 'Ara,

n + 14n2

≤ bn ≤ (n + 1)n2

.

Apì to krit rio twn isosugklinous¸n akolouji¸n, bn → 0.(g) cn = 1√

n2+1+ 1√

n2+2+ · · ·+ 1√

n2+n. 'Opwc sto (b), deÐxte ìti

n√n2 + n

≤ cn ≤ n√n2 + 1

.

Apì to krit rio twn isosugklinous¸n akolouji¸n, cn → 1.

(d) dn = n2n . Parathr ste ìti

dn+1

dn=

n + 12n

→ 12

< 1.

Apì thn 'Askhsh 8, dn → 0.(e) en = 2n+n

3n−n . MporeÐte na gr�yete

en =(2/3)n + (n/3n)

1− (n/3n)→ 0,

afoÔ (2/3)n → 0 kai n/3n → 0.(z) gn = n5

n! . Parathr ste ìti

gn+1

gn=

(1 +

1n

)5 1n + 1

→ 15 · 0 = 0 < 1.

'Ara, gn → 0.(h) hn = nn


hn+1

hn=

(n + 1)n+1n!nn(n + 1)!

=(

1 +1n

)n

→ e > 1.

'Ara, hn → +∞. 'Alloc trìpoc: parathr ste ìti

hn =nn

n!=

n

1· n

2· n

3· · · n

n≥ n · 1 · 1 · · · 1 = n.

(j) xn = 1+22+33+···+nn

nn . Parathr ste ìti

xn =n∑

k=1

kk

nn=

n∑

k=1

(k

n

)k 1nn−k

≤n∑

k=1

1nn−k

.

'Omwc,n∑

k=1

1nn−k

= 1 +1n

+ · · ·+(

1n

)n−1

=1− (

1n

)n

1− 1n

.

'Ara,

1 ≤ xn ≤ n

n− 1

(1− 1

nn

).

Apì to krit rio twn isosugklinous¸n akolouji¸n, xn → 1.

12. Apì ton basikì qarakthrismì tou supremum, gia k�je ε > 0 up�rqeix = x(ε) ∈ A ¸ste a − ε < x ≤ a. Efarmìzontac diadoqik� to parap�nw giaε = 1, 1

2 , 13 , . . ., mporeÐte na breÐte akoloujÐa (an) stoiqeÐwn tou A me a − 1

n <

an ≤ a. Apì to krit rio twn isosugklinous¸n akolouji¸n, an → a.Ac upojèsoume, epiplèon, ìti o a = sup A den eÐnai stoiqeÐo tou A. Up�rqei

a1 ∈ A pou ikanopoieÐ thn a − 1 < a1 ≤ a. 'Omwc, a1 6= a (diìti a /∈ A), �raa− 1 < a1 < a.

Ac upojèsoume ìti èqoume breÐ a1, . . . , am ∈ A pou ikanopoioÔn ta ex c:

(i) a1 < a2 < · · · < am < a.

(ii) Gia k�je k = 1, . . . , m isqÔei a− 1k < ak < a.

Tìte, o sm = max{a− 1m+1 , am} eÐnai mikrìteroc apì ton a. MporoÔme loipìn

na broÔme am+1 ∈ A pou ikanopoieÐ thn sm < am+1 < a (ejhg;hste giatÐ). 'Ara,am < am+1 kai a− 1

m+1 < am+1 < a.Epagwgik�, orÐzetai gnhsÐwc aÔxousa akoloujÐa (an) stoiqeÐwn tou A pou

ikanopoioÔn thn a− 1n < an < a. Apì to krit rio twn isosugklinous¸n akolou-

ji¸n, an → a.

13. 'Estw x ∈ R. Up�rqei q1 ∈ Q pou ikanopoieÐ thn x − 1 < q1 < x (apì thnpuknìthta tou Q sto R).

Ac upojèsoume ìti èqoume breÐ rhtoÔc arijmoÔc q1, . . . , qm pou ikanopoioÔnta ex c:

(i) q1 < q2 < · · · < qm < x.

(ii) Gia k�je k = 1, . . . , m isqÔei x− 1k < qk < x.

Tìte, o sm = max{x − 1m+1 , qm} eÐnai mikrìteroc apì ton x. Lìgw thc puknì-

thtac twn rht¸n stouc pragmatikoÔc arijmoÔc, mporoÔme na broÔme qm+1 ∈ Qsto anoiktì di�sthma (sm, x). Tìte, qm < qm+1 kai x− 1

m+1 < qm+1 < x.Epagwgik�, orÐzetai gnhsÐwc aÔxousa akoloujÐa (qn) rht¸n arijm¸n pou ika-

nopoioÔn thn x− 1n < qn < x. Apì to krit rio twn isosugklinous¸n akolouji¸n,

qn → x.

14. DeÐxte diadoqik� ta ex c:

(i) H (an) orÐzetai kal�. ArkeÐ na deÐxete ìti an ≥ 0 gia k�je n ∈ N. DeÐxteto me epagwg .

(ii) H (an) eÐnai aÔxousa (me epagwg ). Parathr ste ìti an am ≤ am+1 tìte

am+1 =√

2 +√

am ≤√

2 +√

am+1 = am+2.

(iii) H (an) eÐnai �nw fragmènh (me epagwg ). Parathr ste ìti an am ≤ 2 tìteam+1 =

√2 +

√am ≤

√2 +

√2 <

√2 + 2 = 2.

AfoÔ h (an) eÐnai aÔxousa kai �nw fragmènh, sugklÐnei.

15. Parathr ste pr¸ta ìti, an h (an) sugklÐnei tìte to ìrio thc ja ikanopoieÐthn x = 2x+1

x+1 (exhg ste giatÐ). 'Ara x = 1+√

52 x = 1−√5

2 . DeÐxte diadoqik�ta ex c:

(i) H (an) orÐzetai kal�. ArkeÐ na deÐxete ìti an 6= −1 gia k�je n ∈ N. DeÐxteme epagwg ìti an > 0 gia k�je n ∈ N.


am+2 − am+1 =am+1 − am

(am+1 + 1)(am + 1)≥ 0.

(iii) H (an) eÐnai �nw fragmènh (me epagwg ). Parathr ste ìti

am+1 =2am + 1am + 1

≤ 2am + 2am + 1

= 2

gia k�je m. Ja mporoÔsate epÐshc na deÐxete ìti am ≤ 1+√

52 gia k�je

m (afoÔ, apì ta parap�nw, autì eÐnai to {upoy fio ìrio} thc aÔxousacakoloujÐac (an)).

AfoÔ h (an) eÐnai aÔxousa kai �nw fragmènh, sugklÐnei (ston 1+√

52 ).


(i) H (xn) orÐzetai kal�. ArkeÐ na deÐxete ìti xn 6= 0 gia k�je n ∈ N. DeÐxteme epagwg ìti xn > 0 gia k�je n.

(ii) Gia k�je n ≥ 2 isqÔei xn ≥√

a (me epagwg ). Parathr ste ìti

xn+1 =12

(xn +

a

xn

)≥

√xn · a

xn=√

a

gia k�je n ∈ N.(iii) Gia k�je n ≥ 2 isqÔei xn ≥ xn+1 (me epagwg ). Parathr ste ìti

xn − xn+1 =x2

n − a

2xn≥ 0

gia k�je n ∈ N.AfoÔ h (xn)n≥2 eÐnai fjÐnousa kai k�tw fragmènh, sugklÐnei. To ìrio x eÐnaijetikì (apì ta prohgoÔmena èqoume x ≥ √

a) kai prèpei na ikanopoieÐ thn x =12

(x + a

x

), dhlad x2 = a. 'Ara, x =

√a.

17. K�noume pr¸ta thn epiplèon upìjesh ìti a = 0 kai deÐqnoume ìti bn → 0.JewroÔme ε > 0 kai brÐskoume n1(ε) ∈ N me thn idiìthta: gia k�je n ≥ n1 isqÔei|an| < ε/2. Tìte, gia k�je n > n1 èqoume

|bn| ≤ |a1 + · · ·+ an1 |n

+n− n1

n

ε

2<|a1 + · · ·+ an1 |

n+

ε

2.

O arijmìc A := |a1 + · · ·+ an1 | exart�tai apì to ε (afoÔ o n1 exart�tai apì toε) ìqi ìmwc apì to n. Apì thn Arqim deia idiìthta, up�rqei n2(A) = n2(ε) ∈ Nme thn idiìthta: gia k�je n ≥ n2 èqoume

|a1 + · · ·+ an1 |n

=A

n<

ε

2.

An loipìn p�roume n0 = max{n1, n2} tìte, gia k�je n ≥ n0 isqÔei h

|bn| ≤ A

n+

ε

2< ε.

Me b�sh ton orismì, bn → 0.

Gia th genik perÐptwsh, jewr ste thn akoloujÐa a′n := an − a. Tìte,a′n → 0. 'Ara,

bn − a =a1 + · · ·+ an

n− a =

(a1 − a) + · · ·+ (an − a)n

=a′1 + · · ·+ a′n

n→ 0.

'Epetai ìti bn → a.

18. Ac upojèsoume ìti èqoume breÐ fusikoÔc k1 < · · · < km pou ikanopoioÔn taex c:

0 < akm< akm−1 < · · · < ak1 kai 0 < akm

<1m

.

Jètoume εm+1 = min{a1, . . . , akm

, 1m+1

}> 0. AfoÔ inf(A) = inf{an : n ∈

N} = 0 kai εm+1 > 0, up�rqei km+1 ∈ N tètoioc ¸ste akm+1 < εm+1. Apì tonorismì tou εm+1 èpetai ìti (exhg ste giatÐ):

(a) km < km+1, (b) akm+1 < akm , (g) akm+1 <1

m + 1.

'Etsi orÐzetai epagwgik� mia gnhsÐwc fjÐnousa upakoloujÐa (akn) thc (an) hopoÐa sugklÐnei sto 0.

19. Upojètoume ìti oi upakoloujÐec (a2k) kai (a2k−1) sugklÐnoun ston a.'Estw ε > 0. Up�rqei n1 ∈ N me thn idiìthta: gia k�je k ≥ n1 isqÔei

|a2k − a| < ε. EpÐshc, up�rqei n2 ∈ N me thn idiìthta: gia k�je k ≥ n2 isqÔei|a2k−1 − a| < ε. An jèsoume n0 = max{2n1, 2n2 − 1} tìte gia k�je n ≥ n0

isqÔei |an − a| < ε.[Pr�gmati, parathr ste ìti an o n eÐnai �rtioc tìte n = 2k gia k�poion k ≥ n1

en¸ an o n eÐnai perittìc tìte n = 2k − 1 gia k�poion k ≥ n2.]AfoÔ to ε > 0 tan tuqìn, èpetai ìti an → a.

To antÐstrofo eÐnai aplì: èqoume dei ìti an mia akoloujÐa (an) sugklÐneiston a ∈ R tìte k�je upakoloujÐa (akn) thc (an) sugklÐnei ston a.

20. Ac upojèsoume ìti a2k → x, a2k−1 → y kai a3k → z. Parathr ste ìti:(i) H (a6k) eÐnai tautìqrona upakoloujÐa thc (a2k) kai upakoloujÐa thc (a3k).'Ara, h (a6k) sugklÐnei kai x = lim

k→∞a2k = lim

k→∞a6k = lim

k→∞a3k = z.

(ii) H (a6k−3) eÐnai tautìqrona upakoloujÐa thc (a2k−1) kai upakoloujÐa thc(a3k). 'Ara, h (a6k−3) sugklÐnei kai y = lim



k→∞a3k =

z.'Epetai ìti x = y = z. AfoÔ oi (a2k) kai (a2k−1) èqoun to Ðdio ìrio, h 'Askhsh19 deÐqnei ìti h (an) sugklÐnei.

21. DeÐxte diadoqik� ìti:

(a) H (a2n) eÐnai aÔxousa kai �nw fragmènh apì ton a1 en¸ h (a2n−1) eÐnaifjÐnousa kai k�tw fragmènh apì ton a2.

(b) Up�rqoun a, b ∈ R tètoioi ¸ste a2n → a kai a2n−1 → b.

(g) a = b.

(d) an → a = b.

(e) a2n ≤ a ≤ a2n−1 gia k�je n ∈ N.

23. DeÐxte tic isodunamÐec (1) ⇐⇒ (2) ⇐⇒ (3) ⇐⇒ (4):

(1) H akoloujÐa (an) den sugklÐnei ston a.

(2) Up�rqei ε > 0 ¸ste �peiroi ìroi am thc (an) na ikanopoioÔn thn |am−a| ≥ε.

(3) Up�rqoun ε > 0 kai fusikoÐ arijmoÐ k1 < k2 < · · · < kn < · · · ¸ste: giak�je n ∈ N isqÔei |akn

− a| ≥ ε.

(4) Up�rqoun ε > 0 kai upakoloujÐa (akn) thc (an) ¸ste: gia k�je n ∈ NisqÔei |akn − a| ≥ ε.

24. (=⇒) Upojèste pr¸ta ìti an → a. An (akn) eÐnai mia upakoloujÐa thc (an)tìte akn → a. 'Ara, ìlec oi upakoloujÐec thc (akn) sugklÐnoun ki autèc ston a.

(⇐=) Me apagwg se �topo: upojèste ìti h (an) den sugklÐnei ston a. Apìthn 'Askhsh 23, up�rqoun ε > 0 kai upakoloujÐa (akn) thc (an) ¸ste: gia k�jen ∈ N isqÔei |akn − a| ≥ ε. Tìte, an p�roume opoiad pote upakoloujÐa thc(akn), ìloi oi ìroi thc ja eÐnai ε-makri� apì ton a. Dhlad , h (akn) den èqeiupakoloujÐa pou na sugklÐnei ston a.

25. Parathr ste ìti: gia k�je n ∈ N,1

n + 1+

1n + 2

+ · · ·+ 12n

≥ 12n

+ · · ·+ 12n

=n

2n=

12.

Ac upojèsoume ìti h akoloujÐa an = 1 + 12 + · · · + 1

n eÐnai akoloujÐa Cauchy.Tìte, paÐrnontac ε = 1

4 > 0, mporoÔme na broÔme n0 ∈ N me thn idiìthta: giak�je m, n ≥ n0 isqÔei |am − an| < 1

4 .P�rte n ≥ n0 kai m = 2n ≥ n0. Tìte, |a2n − an| < 1

4 . Autì den mporeÐ naisqÔei, diìti

|a2n − an| =(

1 +12

+ · · ·+ 1n

+1

n + 1+ · · ·+ 1

2n

)−

(1 +

12

+ · · ·+ 1n

)

=1

n + 1+

1n + 2

+ · · ·+ 12n

≥ 12.

Exhg ste t¸ra ta ex c:

(a) AfoÔ h (an) den eÐnai akoloujÐa Cauchy, h (an) den sugklÐnei se pragma-tikì arijmì.

(b) AfoÔ h (an) eÐnai aÔxousa kai den sugklÐnei, h (an) den eÐnai �nw fragmènh.

(g) AfoÔ h (an) eÐnai aÔxousa kai den eÐnai �nw fragmènh, anagkastik� an →+∞.

26. Apì thn |an+1 − an| ≤ µ|an − an−1| èpetai (exhg ste giatÐ) ìti: gia k�jen ≥ 2 isqÔei

|an+1 − an| ≤ µn−1|a2 − a1| = |b− a| · µn−1.

An loipìn m,n ∈ N kai m > n, tìte

|am − an| ≤ |am − am−1|+ · · ·+ |an+1 − an|≤ |b− a| (µn−1 + · · ·+ µm−2

)

= |b− a| · µn−1 · 1− µm−n

1− µ

≤ |b− a|µn−1

1− µ

=|b− a|

µ(1− µ)· µn.

Jewr ste ε > 0 kai breÐte n0 ∈ N pou ikanopoieÐ thn |b−a|µ(1−µ)µ

n0 < ε. Tètoiocn0 up�rqei giatÐ µn → 0 ìtan n →∞. Tìte, an m > n ≥ n0,

|am − an| ≤ |b− a|µ(1− µ)

· µn ≤ |b− a|µ(1− µ)

· µn0 < ε.

Dhlad , h (an) eÐnai akoloujÐa Cauchy.

27. Parathr ste ìti: gia k�je n ≥ 2 isqÔei

an+1 − an =an + an−1

2− an = −an − an−1

2,

dhlad

|an+1 − an| ≤ 12|an − an−1|.

Apì thn 'Askhsh 26, h (an) eÐnai akoloujÐa Cauchy.

28. 'Estw ε > 0 kai èstw n0 ∈ N. DeÐxte ìti ta ex c eÐnai isodÔnama:

(a) Gia k�je m,n ≥ n0 isqÔei |am − an| ≤ ε.

(b) Gia k�je m > n ≥ n0 isqÔei |am − an| ≤ ε.

(g) Gia k�je n ≥ n0 kai k�je k ∈ N isqÔei |an+k − an| ≤ ε.

(d) Gia k�je n ≥ n0 isqÔei bn := sup{|an+k − an| : k ∈ N} ≤ ε.

Qrhsimopoi¸ntac thn isodunamÐa twn (a) kai (d) deÐxte ìti h (an) eÐnai akoloujÐaCauchy (isodÔnama, sugklÐnei) an kai mìno an bn → 0.

G. Ask seic

1. (a) OrÐzoume a = max{a1, a2, . . . , ak}. Tìte, gia k�je n ∈ N èqoume an <

an1 + · · ·+ an

k ≤ kan. 'Ara,

a < bn := n√

an1 + an

2 + · · ·+ ank ≤

n√

kan = an√

k.

AfoÔ limn→∞

n√

k = 1 (blèpe basik �skhsh 9), apì to krit rio twn isosugklinou-s¸n akolouji¸n èqoume bn → a. Gia par�deigma,

n√

2n + 3n + 7n → 7.

(b) To pl joc twn prosjetèwn (ston orismì tou n-ostoÔ ìrou) den eÐnai staje-rì. Doulèyte ìmwc ìpwc sto (a): parathr ste ìti nn < 1n+2n+· · ·+nn < n·nn

an n ≥ 2. 'Ara,

1 < xn =1n

n√

1n + 2n + · · ·+ nn < n√

n

gia k�je n ≥ 2. AfoÔ n√

n → 1, efarmìzetai to krit rio twn isosugklinous¸nakolouji¸n, kai xn → 1.

2. Qrhsimopoi ste to gegonìc ìti xn =(1 + 1

n

)n → e. Gia par�deigma,

(a) an =(1 + 1

n−1

)n−1

= xn−1 → e.

(b) bn =(1 + 2

n

)n =(

n+2n+1

)n (n+1

n

)n = n+1n+2xn+1xn → e2.

(g) 1cn

=(

nn−1

)n

= nn−1xn−1 → e, �ra cn → 1

e .

(d) dn =(1− 1

n2

)n =(1− 1

n

)n (1 + 1

n

)n → 1e · e = 1.

(e) e3n =

(1 + 2

3n

)3n → e2 (giatÐ?), �ra en → 3√

e2.


(i) an > 0 kai bn > 0 gia k�je n ∈ N.

(ii) an ≤ bn gia k�je n ∈ N. Apì ton anadromikì orismì (anex�rthta m�listaapì to poioÐ eÐnai oi an kai bn) èqete

an+1 =√

anbn ≤ an + bn

2= bn+1.

(iii) H (an) eÐnai aÔxousa. Parathr ste ìti an+1 =√

anbn ≥√

a2n = an gia

k�je n ∈ N.

(iv) H (bn) eÐnai fjÐnousa. Parathr ste ìti bn+1 = an+bn

2 ≤ 2bn

2 = bn giak�je n ∈ N.

Apì ta parap�nw, h (an) eÐnai aÔxousa kai �nw fragmènh apì ton b1, en¸ h (bn)eÐnai fjÐnousa kai k�tw fragmènh apì ton a1 (exhg ste giatÐ). 'Ara, up�rqouna, b ∈ R ¸ste an → a kai bn → b. Apì thn bn+1 = an+bn

2 èpetai ìti b = a+b2 ,

dhlad a = b.

4. Parathr ste ìti

xn+2 − xn+1 = −xn+1 − xn

3.

'Ara, h akoloujÐa yn = xn+1−xn ikanopoieÐ thn anadromik sqèsh yn+1 = −yn

3 .'Epetai (exhg ste giatÐ) ìti

yn =(−1

3

)n−1

y1 =(−1

3

)n−1

(b− a).

Parathr ste ìti

xn = x1 + (x2 − x1) + · · ·+ (xn − xn−1) = a + y1 + · · ·+ yn−1.

'Ara,

xn = a +

(1 +

(−1

3

)+

(−1

3

)2

+ · · ·+(−1

3

)n−2)

(b− a)

= a +1− (− 1

3 )n−1

1− (− 13 )

(b− a) → a +34(b− a) =

3b + a

4.

5. Upojètoume ìti limn→∞an+1−an

bn+1−bn= λ ìpou λ ∈ R (h perÐptwsh λ = +∞

exet�zetai an�loga). 'Estw ε > 0. Qrhsimopoi¸ntac kai to gegonìc ìti bn →+∞, blèpoume ìti up�rqei n1(ε) ∈ N ¸ste: gia k�je n ≥ n1 isqÔei bn > 0 kai

λ− ε

2<

an+1 − an

bn+1 − bn< λ +

ε

2.

AfoÔ h (bn) eÐnai gnhsÐwc aÔxousa, èqoume bn+1−bn > 0. 'Ara, gia k�je n ≥ n1

isqÔei (λ− ε

2

)(bn+1 − bn) < an+1 − an <

(λ +

ε

2

)(bn+1 − bn).

'Epetai (exhg ste giatÐ) ìti: gia k�je n > n1 isqÔei(λ− ε

2

)(bn − bn1) < an − an1 <

(λ +

ε

2

)(bn − bn1).

Diair¸ntac me bn paÐrnoume

(λ− ε

2

) (1− bn1

bn

)+

an1

bn<

an

bn<

(λ +

ε

2

) (1− bn1

bn

)+

an1

bn.

Parathr ste ìti

limn→∞

[(λ− ε

2

) (1− bn1

bn

)+

an1

bn

]= λ− ε

2

kai

limn→∞

[(λ +

ε

2

) (1− bn1

bn

)+

an1

bn

]= λ +

ε

2

giatÐ bn → +∞ ìtan n → ∞. 'Ara (exhg ste giatÐ) up�rqei n2 ∈ N, pouexart�tai apì to n1 kai apì to ε, ¸ste: gia k�je n ≥ n2 isqÔei

λ− ε <(λ− ε

2

) (1− bn1

bn

)+

an1

bn<

(λ +

ε

2

) (1− bn1

bn

)+

an1

bn< λ + ε.

Sunep¸c, an n ≥ n0 := max{n1, n2}, èqoume

λ− ε <an

bn< λ + ε.

AfoÔ to ε > 0 tan tuqìn, sumperaÐnoume ìti

limn→∞

an

bn= λ.

6. AfoÔ 1an→ 1

a , h basik �skhsh 17 deÐqnei ìti

1bn

=1a1

+ · · ·+ 1an

n→ 1

a.

Ara, bn → a. Gia thn γn, parathr ste ìti bn ≤ γn ≤ δn := a1+···+an

n apì thnanisìthta armonikoÔ-gewmetrikoÔ-arijmhtikoÔ mèsou, kai efarmìste to krit riotwn isosugklinous¸n akolouji¸n se sunduasmì me thn 'Askhsh 17.

7. Parathr ste ìti

an

n=

(an − an−1) + · · ·+ (a2 − a1) + a1

n=

bn−1 + · · ·+ b1

n+

a1

n

ìpou bn := an+1 − an → a. T¸ra, qrhsimopoi ste thn 'Askhsh 17.

8. ArkeÐ na deÐxoume ìti h (an) eÐnai �nw fragmènh. Tìte, h (an) sugklÐnei kai,apì thn 'Askhsh 17, lim

n→∞an = lim

n→∞bn = a.

K�noume pr¸ta thn epiplèon upìjesh ìti a1 ≥ 0. Tìte, an ≥ a1 ≥ 0 giak�je n ∈ N. AfoÔ h (bn) sugklÐnei, eÐnai fragmènh: eidikìtera, up�rqei M ∈ R¸ste: gia k�je k ∈ N isqÔei kbk = a1 + · · ·+ ak ≤ kM . PaÐrnontac k = 2n kaiqrhsimopoi¸ntac thn upìjesh ìti h (an) eÐnai aÔxousa, gr�foume

nan ≤ an+1 + · · ·+ a2n ≤ a1 + · · ·+ a2n = 2nb2n ≤ 2nM.

Dhlad , h (an) eÐnai �nw fragmènh apì ton 2M . PoÔ qrhsimopoi jhke h upìjeshìti oi ìroi thc (an) eÐnai mh arnhtikoÐ?

Gia th genik perÐptwsh, jewr ste thn (aÔxousa akoloujÐa) a′n = an − a1

kai thn b′n := a′1+···+a′nn . Apì thn upìjesh èqoume b′n → a−a1 kai, ìpwc orÐsthke

h (a′n), èqoume a′1 = 0. 'Ara, h (a′n) eÐnai �nw fragmènh. 'Epetai ìti h (an) eÐnai�nw fragmènh (exhg ste giatÐ).

9. DeÐxte pr¸ta ìti h (an) orÐzetai kal� kai an > 0 gia k�je n ∈ N. EpÐshc,deÐxte ìti an an → x tìte x =

√3.

(i) Upojèste ìti 0 < a1 <√

(a) a2 >√

3.

(b) Gia k�je n ∈ N isqÔei h an+2 = 3+2an

2+an.

(g) Gia k�je k ∈ N isqÔoun oi a2k−1 <√

3 kai a2k−1 < a2k+1.

(d) Gia k�je k ∈ N isqÔoun oi a2k >√

3 kai a2k+2 < a2k.

Qrhsimopoi¸ntac thn 'Askhsh 21 deÐxte ìti oi (a2k−1) kai (a2k) sugklÐnoun.Qrhsimopoi¸ntac thn anadromik sqèsh an�mesa ston an+2 kai ton an deÐxte ìtilim

k→∞a2k = lim

k→∞a2k−1 =

√3. Apì thn 'Askhsh 19 èpetai ìti an →

√3.

(ii) Exet�ste me ton Ðdio trìpo thn perÐptwsh a1 >√

3.(iii) Tèloc, deÐxte ìti an a1 =

√3 tìte èqoume an =

√3 gia k�je n ∈ N.




1. L�joc. H akoloujÐa an = (−1)n eÐnai fragmènh all� den sugklÐnei.

2. Swstì. Upojètoume ìti an → a ∈ R. PaÐrnoume ε = 1 > 0. MporoÔme nabroÔme n0 ∈ N ¸ste |an − a| < 1 gia k�je n ≥ n0. Dhlad ,

an n ≥ n0, tìte |an| < 1 + |a|.

JètoumeM = max{|a1|, . . . , |an0 |, 1 + |a|}

kai elègqoume ìti |an| ≤ M gia k�je n ∈ N (diakrÐnete peript¸seic: n ≤ n0 kain > n0). 'Ara, h (an) eÐnai fragmènh.

3. Swstì. Upojètoume ìti h (an) sugklÐnei ston pragmatikì arijmì a. Epilè-goume ε = 1

2 kai brÐskoume n0 ∈ N ¸ste: gia k�je n ≥ n0 isqÔei |an − a| < 12 .

Tìte, an n ≥ n0 èqoume

|an − an0 | = |(an − a) + (a− an0)| ≤ |an − a|+ |a− an0 | <12

+12

= 1.

'Omwc, oi an kai an0 eÐnai akèraioi, �ra an = an0 . Dhlad , h (an) eÐnai telik�stajer : gia k�je n ≥ n0 èqoume an = an0 .

H antÐstrofh kateÔjunsh eÐnai apl : an gia mia akoloujÐa (an) sto R up�r-qoun n0 ∈ N kai a ∈ R ¸ste gia k�je n ≥ n0 na isqÔei an = a, tìte an → a

(exhg ste, me b�sh ton orismì tou orÐou).

4. L�joc. Ac upojèsoume ìti (an) eÐnai mia gnhsÐwc fjÐnousa akoloujÐa fusik¸narijm¸n. To sÔnolo A = {an : n ∈ N} eÐnai mh kenì uposÔnolo tou N. Apì

thn arq thc kal c di�taxhc, èqei el�qisto stoiqeÐo. Dhlad , up�rqei m ∈ N¸ste: gia k�je n ∈ N isqÔei am ≤ an. Autì eÐnai �topo, afoÔ am+1 ∈ A kaiam+1 < am.

5. L�joc. H akoloujÐa an =√

2n eÐnai akoloujÐa �rrhtwn arijm¸n, ìmwc an → 0.

6. Swstì. DiakrÐnete peript¸seic. An o x eÐnai �rrhtoc, mporeÐte na jewr seteth stajer akoloujÐa an = x. An o x eÐnai rhtìc, mporeÐte na jewr sete thnakoloujÐa an = x +

√2

n .

7. Swstì. Upojètoume pr¸ta ìti an → 0. 'Estw M > 0. AfoÔ an → 0,efarmìzontac ton orismì me ε = 1

M > 0, mporoÔme na broÔme n0 = n0(ε) =n0(M) ∈ N ¸ste: gia k�je n ≥ n0 isqÔei 0 < an < 1

M . Dhlad , up�rqein0 = n0(M) ∈ N ¸ste: gia k�je n ≥ n0 isqÔei 1

an> M . 'Epetai ìti 1

an→ +∞.

Gia thn antÐstrofh kateÔjunsh ergazìmaste me an�logo trìpo.

8. L�joc. H akoloujÐa an = (−1)n

n sugklÐnei sto 0 all� den eÐnai monìtonh.

9. Swstì. 'Estw M > 0. AfoÔ h (an) den eÐnai �nw fragmènh, up�rqei n0 ∈ N¸ste an0 > M . AfoÔ h (an) eÐnai aÔxousa, gia k�je n ≥ n0 isqÔei an ≥ an0 >

M . AfoÔ o M > 0 tan tuq¸n, sumperaÐnoume ìti an → +∞.

10. L�joc. H an = (−1)n eÐnai fragmènh kai h bn = 1 sugklÐnei, ìmwc hanbn = (−1)n den sugklÐnei.

11. L�joc. H an = (−1)n den sugklÐnei, ìmwc h |an| = 1 sugklÐnei.

12. Swstì. H basik idèa eÐnai ìti, afoÔ a > 0 kai an → a, ja up�rqei n0 methn idiìthta: gia k�je n ≥ n0 isqÔei an > a/2. Dhlad , telik� ìloi oi ìroi thc(an) xepernoÔn ton jetikì arijmì a/2.

Pr�gmati, an efarmìsete ton orismì tou orÐou gia thn (an) me ε = a/2 > 0,mporeÐte na breÐte n0 ∈ R ¸ste: gia k�je n ≥ n0,

|an − a| < ε = a/2 =⇒ a/2 < an < 3a/2.

Tìte, o jetikìc arijmìc min{a1, . . . , an0 , a/2} eÐnai k�tw fr�gma tou sunìlouA = {an : n ∈ N} (exhg ste giatÐ). Sunep¸c, inf(A) ≥ a/2 > 0.

13. Swstì. H basik idèa eÐnai ìti, afoÔ a1 > 0 kai an → 0, ja up�rqei n0 methn idiìthta: gia k�je n ≥ n0 isqÔei an < a1. Dhlad , up�rqei stoiqeÐo tou A

megalÔtero apì {ìla} (ektìc apì peperasmèna to pl joc) ta stoiqeÐa tou A.Pr�gmati, an efarmìsete ton orismì tou orÐou gia thn (an) me ε = a1 > 0,

mporeÐte na breÐte n0 ∈ R ¸ste: gia k�je n ≥ n0,

an = |an − 0| < ε = a1.

Tìte, o megalÔteroc apì touc a1, . . . , an0 eÐnai to mègisto stoiqeÐo tou A: an keisto A kai eÐnai megalÔteroc Ðsoc apì k�je an (exhg ste giatÐ).

14. L�joc. Jewr ste thn akoloujÐa (an) me a2k = k kai a2k−1 = 1 gia k�jek ∈ N. Tìte, an > 0 kai h (an) den eÐnai �nw fragmènh, ìmwc an → +∞ (anautì Ðsque, ja èprepe ìloi telik� oi ìroi thc (an) na eÐnai megalÔteroi apì 2, toopoÐo den isqÔei afoÔ ìloi oi perittoÐ ìroi thc eÐnai Ðsoi me 1).

15. L�joc. Qrhsimopoi ste to par�deigma thc prohgoÔmenhc er¸thshc gia nadeÐxete ìti mporeÐ na isqÔei h prìtash {gia k�je M > 0 up�rqoun �peiroi ìroithc (an) pou eÐnai megalÔteroi apì M} qwrÐc na isqÔei h prìtash {an → +∞}.

H �llh kateÔjunsh eÐnai swst : an an → +∞ tìte (apì ton orismì) giak�je M > 0 ìloi telik� oi ìroi thc (an) eÐnai megalÔteroi apì M . 'Ara, giak�je M > 0 up�rqoun �peiroi ìroi thc (an) pou eÐnai megalÔteroi apì M .

16. Swstì. Upojètoume pr¸ta ìti up�rqei upakoloujÐa (akn) thc (an) ¸ste

akn→ +∞. 'Estw M > 0. AfoÔ akn

→ +∞, ìloi telik� oi ìroi akneÐnai mega-

lÔteroi apì M . Dhlad , up�rqoun �peiroi ìroi thc (an) pou eÐnai megalÔteroiapì M . AfoÔ o M > 0 tan tuq¸n, h (an) den eÐnai �nw fragmènh.

AntÐstrofa, ac upojèsoume ìti h (an) den eÐnai �nw fragmènh. Ja broÔme e-pagwgik� k1 < · · · < kn < kn+1 < · · · ¸ste akn > n. Tìte, gia thn upakoloujÐa(akn) thc (an) ja èqoume akn → +∞:

AfoÔ h (an) den eÐnai �nw fragmènh, mporoÔme na broÔme k1 ∈ N ¸steak1 > 1. Ac upojèsoume ìti èqoume breÐ k1 < · · · < km ¸ste akj > j gia k�jej = 1, . . . , m. Jètoume M = max{a1, a2, . . . , akm , m + 1}. AfoÔ h (an) deneÐnai �nw fragmènh, mporoÔme na broÔme km+1 ∈ N ¸ste akm+1 > M . Apì tonorismì tou M èqoume akm+1 > m + 1 kai akm+1 > an gia k�je n = 1, 2, . . . , km.Sunep¸c, km+1 > km. Autì oloklhr¸nei to epagwgikì b ma.

17. Swstì. Upojètoume ìti an → a. 'Estw (bn) = (akn) mia upakoloujÐa thc(an) kai èstw ε > 0. Up�rqei n0 ∈ N ¸ste: gia k�je n ≥ n0 isqÔei |an−a| < ε.Parathr ste ìti kn0 ≥ n0 (diìti h (kn) eÐnai gnhsÐwc aÔxousa akoloujÐa fusik¸narijm¸n). Tìte, gia k�je n ≥ n0 èqoume kn ≥ n0, �ra |bn − a| = |akn − a| < ε.'Epetai ìti bn → a.

18. Swstì. JumhjeÐte ton orismì tou shmeÐou koruf c miac akoloujÐac (an):h (an) èqei shmeÐo koruf c ton ìro ak an ak ≥ am gia k�je m ≥ k.

AfoÔ h (an) den èqei fjÐnousa upakoloujÐa, den mporeÐ na èqei �peira shmeÐakoruf c: pr�gmati, ac upojèsoume ìti up�rqoun k1 < k2 < · · · < kn < · · · ¸steoi ìroi ak1 , . . . , akn , . . . na eÐnai shmeÐa koruf c thc (an). Tìte,

ak1 ≥ ak2 ≥ · · · ≥ akn ≥ · · · ,

dhlad h upakoloujÐa (akn) eÐnai fjÐnousa.'Ara, h (an) èqei peperasmèna to pl joc shmeÐa koruf c: Up�rqei dhlad

k1 ∈ N (to teleutaÐo shmeÐo koruf c o k1 = 1 an den up�rqoun shmeÐa koruf c)me thn idiìthta: an n ≥ k1, up�rqei n′ > n ¸ste an′ > an.

BrÐskoume k2 > k1 ¸ste ak1 < ak2 , katìpin brÐskoume k3 > k2 ¸ste ak2 >

ak3 kai oÔtw kajex c. Up�rqoun dhlad k1 < k2 < · · · < kn < · · · ¸ste

ak1 < ak2 < · · · < akn < · · · .

'Ara, h (an) èqei toul�qiston mÐa gnhsÐwc aÔxousa upakoloujÐa.

19. L�joc. Jewr ste thn akoloujÐa (an) me a2k = k kai a2k−1 = 1 gia k�jek ∈ N. Tìte, an → 1, ìmwc h (an) den èqei upakoloujÐa pou na sugklÐnei sek�poion pragmatikì arijmì b 6= 1 (exhg ste giatÐ).ShmeÐwsh. An k�noume thn epiplèon upìjesh ìti h (an) eÐnai fragmènh, tìte hprìtash eÐnai swst . Sthn 'Askhsh 23 parak�tw apodeiknÔetai ìti an h akolou-jÐa (an) den sugklÐnei ston a tìte up�rqoun ε > 0 kai upakoloujÐa (akn) thc(an) ¸ste: gia k�je n ∈ N isqÔei |akn

− a| ≥ ε. AfoÔ h (an) eÐnai fragmènh, h(akn

) eÐnai epÐshc fragmènh. Apì to je¸rhma Bolzano–Weierstrass h (akn) èqei

upakoloujÐa (akln) h opoÐa sugklÐnei se k�poion b ∈ R. 'Omwc, |akln

− a| ≥ ε

gia k�je n ∈ N, �ra |b− a| = limn→∞ |akln− a| ≥ ε. Dhlad , b 6= a.

20. L�joc. Apì to je¸rhma Bolzano–Weierstrass, k�je fragmènh akoloujÐaèqei sugklÐnousa upakoloujÐa.

21. L�joc. Jewr ste thn akoloujÐa (an) me a2k = k kai a2k−1 = 1 gia k�jek ∈ N. H (an) den eÐnai fragmènh, ìmwc h upakoloujÐa (a2k−1) eÐnai stajer (�ra, fragmènh).

22. Swstì. Jewr ste mia upakoloujÐa (akn) thc (an). An n ∈ N tìte kn <

kn+1. H (an) eÐnai aÔxousa, �ra: an s, t ∈ N kai s < t tìte as ≤ at (exhg stegiatÐ). PaÐrnontac s = kn kai t = kn+1 sumperaÐnoume ìti akn ≤ akn+1 .

23. Swstì. Upojètoume ìti h (an) eÐnai aÔxousa kai ìti up�rqei upakoloujÐa(akn) thc (an) h opoÐa sugklÐnei ston a ∈ R.

AfoÔ akn → a, h (akn) eÐnai fragmènh. Sunep¸c, up�rqei M ∈ R tètoioc¸ste: gia k�je n ∈ N isqÔei akn ≤ M . Ja deÐxoume ìti: gia k�je n ∈ N isqÔeian ≤ M . Tìte, h (an) eÐnai aÔxousa kai �nw fragmènh, �ra sugklÐnei.

Jewr ste tuqìnta n ∈ N. AfoÔ n ≤ kn kai h (an) eÐnai aÔxousa, èqoumean ≤ akn ≤ M .

24. Swstì. Ja broÔme epagwgik� k1 < · · · < kn < kn+1 < · · · ¸ste |akn | < 1n3 .

Tìte, gia thn upakoloujÐa (akn) thc (an) ja èqoume n2akn → 0 (exhg ste giatÐ).AfoÔ h an → 0, mporoÔme na broÔme k1 ∈ N ¸ste |ak1 | < 1. Ac upojèsoume

ìti èqoume breÐ k1 < · · · < km ¸ste |akj | < 1j3 gia k�je j = 1, . . . , m. Jètoume

ε = 1(m+1)3 > 0. AfoÔ h an → 0, mporoÔme na broÔme n0 ∈ N ¸ste |an| < 1

(m+1)3

gia k�je n ≥ n0. Eidikìtera, up�rqei km+1 > km ¸ste |akm+1 | < 1(m+1)3 . Autì

oloklhr¸nei to epagwgikì b ma.

25. Swstì. Jètoume A = {an : n ∈ N}. Apì ton basikì qarakthrismì tousupremum, gia k�je ε > 0 up�rqei x = x(ε) ∈ A ¸ste 1 − ε < x ≤ 1. AfoÔ1 /∈ A, èqoume thn isqurìterh anisìthta 1− ε < x < 1.

Qrhsimopoi¸ntac to parap�nw, ja broÔme epagwgik� k1 < · · · < kn < kn+1 <

· · · ¸ste ak1 < · · · < akn < akn+1 < · · · kai 1 − 1n < akn < 1. Tìte, gia thn

gnhsÐwc aÔxousa upakoloujÐa (akn) thc (an) ja èqoume akn → 1.

Efarmìzontac ton qarakthrismì tou supremum me ε = 1, brÐskoume ak1 ∈ A

pou ikanopoieÐ thn 0 < ak1 < 1.Ac upojèsoume ìti èqoume breÐ k1 < · · · < km ¸ste ak1 < · · · < akm

kai 1−1j < akj

< 1 gia k�je j = 1, . . . , m. Jètoume s = max{1− 1m+1 , a1, a2, . . . , akm

}.AfoÔ s < 1, mporoÔme loipìn na broÔme akm+1 ∈ A pou ikanopoieÐ thn s <

akm+1 < 1. Tìte, km+1 > km, akm< akm+1 kai 1 − 1

m+1 < akm+1 < 1. Autìoloklhr¸nei to epagwgikì b ma.

26. Swstì. An an → a tìte oi upakoloujÐec (a2k) kai (a2k−1) sugklÐnoun stona. Apì thn upìjesh ìti an+2 = an gia k�je n ∈ N, blèpoume ìti oi (a2k) kai(a2k−1) eÐnai stajerèc akoloujÐec: up�rqoun x, y ∈ R ¸ste

x = a1 = a3 = a5 = · · · kai y = a2 = a4 = a6 = · · · .

Apì tic a2k−1 → x kai a2k → y èpetai ìti x = y = a. 'Ara, h (an) eÐnai stajer .


1. (a) AfoÔ n + (−1)n ≥ n − 1, h anisìthta n + (−1)n > 103 isqÔei ann ≥ n0 := 103 + 2. Dhlad , isqÔei telik�.

(b) An o n eÐnai �rtioc kai megalÔteroc apì 1000, tìte (−1)nn = n > 1000. 'Ara,h (−1)nn > 1000 isqÔei gia �peirouc to pl joc fusikoÔc n. An o n eÐnai perittìc,tìte h (−1)nn > 1000 den isqÔei (exhg ste giatÐ). 'Ara, h (−1)nn > 1000 denisqÔei telik�.

(g) 'Eqoume 1n + (−1)n

n2 ≤ 1n + 1

n2 ≤ 2n . An loipìn n > 2 · 1010, tìte 1

n + (−1)n

n2 <

10−10 (dhlad , h anisìthta isqÔei telik�).

(d) H (1+(−1)n)n < 2 isqÔei an kai mìno an o n eÐnai perittìc (exhg ste giatÐ).Sunep¸c, isqÔei gia �peirouc to pl joc fusikoÔc n, ìqi ìmwc telik�.

(e) An n2 + 2n < 106 tìte n2 < 106, dhlad n < 1000. 'Ara, h anisìthta denisqÔei gia �peirouc to pl joc fusikoÔc n (oÔte telik�).

2.(a) PaÐrnontac ε = 0.001 > 0 brÐskoume n1 ∈ N ¸ste: gia k�je n ≥ n1

isqÔei 1.999 < an < 2.001. 'Ara, k�je n ≥ n1 an kei sto A1. To N \ A1 eÐnaipeperasmèno (kai to A1 �peiro, kai m�lista, telikì tm ma tou N).(g) PaÐrnontac ε = 0.02 > 0 brÐskoume n3 ∈ N ¸ste: gia k�je n ≥ n3 isqÔei1.98 < an < 2.02. 'Ara, k�je n ≥ n3 an kei sto N\A3. To A3 eÐnai peperasmèno(kai to N \A3 �peiro, kai m�lista, telikì tm ma tou N).(e) PaÐrnontac ε = 0.00003 > 0 brÐskoume n5 ∈ N ¸ste: gia k�je n ≥ n5 isqÔei1.99997 < an < 2.00003 < 2.0001. 'Ara, k�je n ≥ n5 an kei sto A5. To N \A5

eÐnai peperasmèno (kai to A5 �peiro, kai m�lista, telikì tm ma tou N).(st) To A6 mporeÐ na eÐnai peperasmèno �peiro, exart�tai apì thn (an). P�rtesan paradeÐgmata tic akoloujÐec

an = 2, an = 2 +1n

, an = 2 +(−1)n

n.

'Olec ikanopoioÔn thn limn→∞

an = 2. Sthn pr¸th perÐptwsh èqoume A6 = N kaiN \A6 = ∅. Sthn deÔterh, A6 = ∅ kai N \A6 = N. Sthn trÐth, tìso to A6 ìsokai to N \ A6 eÐnai �peira sÔnola (to sÔnolo twn peritt¸n kai to sÔnolo twn�rtiwn fusik¸n, antÐstoiqa).

3. Ac doÔme gia par�deigma thn (cn): apì tic 2n = (1 + 1)n ≥ 1 + n > n kain2 + 1 ≥ n + 1 > n èqoume 0 < cn < 1

n gia k�je n ∈ N. Pr�gmati, an on eÐnai thc morf c n = 3k + 1 gia k�poion mh arnhtikì akèraio k (dhlad , ann = 1, 4, 7, 10, 13, . . .) tìte 0 < cn = 1

2n < 1n . An p�li n 6= 3k + 1 gia k�je mh

arnhtikì akèraio k, tìte 0 < cn = 1n2+1 < 1

n .An t¸ra mac d¸soun (tuqìn) ε > 0, up�rqei n0 = n0(ε) ∈ N ¸ste 1

n0< ε.

Tìte, gia k�je n ≥ n0 èqoume

−ε < 0 < cn <1n≤ 1

n0< ε,

dhlad , |cn| < ε. Sunep¸c, cn → 0.

4. Parathr ste ìti

|an − 1| =∣∣∣∣n2 − n

n2 + n− 1

∣∣∣∣ =∣∣∣∣−2n

n2 + n

∣∣∣∣ =2n

n2 + n<

2n

n2=

2n

.

Gia tuqìn ε > 0 breÐte n0(ε) ∈ N ¸ste 2n0

< ε. Tìte, gia k�je n ≥ n0 isqÔei|an − 1| < 2

n ≤ 2n0

< ε.

5. Epilègoume ε = a/2 > 0. AfoÔ limn→∞

an = a, up�rqei n0(ε) ∈ N me thnidiìthta: gia k�je n ≥ n0 isqÔei |an − a| < a/2. IsodÔnama, gia k�je n ≥ n0

isqÔei a/2 < an < 3a/2. 'Ara, an > a/2 > 0 telik� (apì ton n0-ostì ìro kaipèra).

6. (a) Ac upojèsoume ìti b < a. Tìte, an − bn → a− b > 0. Apì thn 'Askhsh5, up�rqei n0 ∈ N me thn idiìthta: gia k�je n ≥ n0 isqÔei

an − bn >a− b

2> 0.

'Atopo, afoÔ an ≤ bn gia k�je n ∈ N (apì thn upìjesh).(b) 'Oqi: an orÐsoume an = 0 kai bn = 1

n , tìte an < bn gia k�je n ∈ N, all�lim

n→∞an = lim

n→∞bn = 0. Ja èqoume ìmwc a ≤ b (apì to pr¸to er¸thma).

(g) Jewr ste tic stajerèc akoloujÐec γn = m, δn = M kai efarmìste to (a).

8. (g) Jètoume ε = `−12 > 0. AfoÔ an+1

an→ `, up�rqei n0 ∈ N ¸ste: gia k�je

n ≥ n0,an+1

an> `− ε =

` + 12

.




θn0· θn.

AfoÔ limn→∞


(d) Jètoume ε = 1−`2 > 0. AfoÔ

∣∣∣an+1an


an+1

an

∣∣∣∣ < ` + ε =` + 1

2.



|an| < ρn−n0 |an0 | =|an0 |ρn0

· ρn.

AfoÔ limn→∞

ρn = 0, èpetai ìti an → 0.

10. (a) H (|bn|) eÐnai fjÐnousa kai k�tw fragmènh apì to 0. 'Ara, up�rqei x ≥ 0¸ste |bn| → x. Apì thn |bn+1| = |bn| · |a|, paÐrnontac ìrio wc proc n kai stadÔo mèlh, èqoume x = x · |a|. AfoÔ |a| 6= 1, sumperaÐnoume ìti x = 0.(b) Parathr ste ìti

∣∣∣∣1− x2

1 + x2

∣∣∣∣ =|1− x2|1 + x2

≤ 1 + x2

1 + x2= 1.

EpÐshc, an x 6= 0 tìte 1−x2

1+x2 6= ±1 (exhg ste giatÐ), �ra∣∣∣∣1− x2

1 + x2

∣∣∣∣ < 1.

Apì to (a), an x 6= 0 tìte h akoloujÐa(

1−x2

1+x2

)n

→ 0. Tèloc, an x = 0 èqoume(1−x2

1+x2

)n

= 1 → 1.

Dhlad , h akoloujÐa(

1−x2

1+x2

)n

sugklÐnei, ìpoio ki an eÐnai to x.

11. (a) an = n3+5n2+22n3+9 . MporeÐte na gr�yete

an =1 + 5

n + 2n3

2 + 9n3

→ 12.

(b) bn = 1n2 + 1

(n+1)2 + · · · + 1(2n)2 . Parathr ste ìti 1

(2n)2 ≤ 1(n+k)

2 ≤ 1n2 gia

k�je k = 0, . . . , n. 'Ara,

n + 14n2

≤ bn ≤ (n + 1)n2

.

Apì to krit rio twn isosugklinous¸n akolouji¸n, bn → 0.(g) cn = 1√

n2+1+ 1√

n2+2+ · · ·+ 1√

n2+n. 'Opwc sto (b), deÐxte ìti

n√n2 + n

≤ cn ≤ n√n2 + 1

.

Apì to krit rio twn isosugklinous¸n akolouji¸n, cn → 1.(d) dn = n

2n . Parathr ste ìti

dn+1

dn=

n + 12n

→ 12

< 1.

Apì thn 'Askhsh 8, dn → 0.(e) en = 2n+n

3n−n . MporeÐte na gr�yete

en =(2/3)n + (n/3n)

1− (n/3n)→ 0,

afoÔ (2/3)n → 0 kai n/3n → 0.(z) gn = n5


gn+1

gn=

(1 +

1n

)5 1n + 1

→ 15 · 0 = 0 < 1.

'Ara, gn → 0.(h) hn = nn


hn+1

hn=

(n + 1)n+1n!nn(n + 1)!

=(

1 +1n

)n

→ e > 1.

'Ara, hn → +∞. 'Alloc trìpoc: parathr ste ìti

hn =nn

n!=

n

1· n

2· n

3· · · n

n≥ n · 1 · 1 · · · 1 = n.

(j) xn = 1+22+33+···+nn

nn . Parathr ste ìti

xn =n∑

k=1

kk

nn=

n∑

k=1

(k

n

)k 1nn−k

≤n∑

k=1

1nn−k

.

'Omwc,n∑

k=1

1nn−k

= 1 +1n

+ · · ·+(

1n

)n−1

=1− (

1n

)n

1− 1n

.

'Ara,

1 ≤ xn ≤ n

n− 1

(1− 1

nn

).

Apì to krit rio twn isosugklinous¸n akolouji¸n, xn → 1.

12. Apì ton basikì qarakthrismì tou supremum, gia k�je ε > 0 up�rqeix = x(ε) ∈ A ¸ste a − ε < x ≤ a. Efarmìzontac diadoqik� to parap�nw giaε = 1, 1

2 , 13 , . . ., mporeÐte na breÐte akoloujÐa (an) stoiqeÐwn tou A me a − 1

n <

an ≤ a. Apì to krit rio twn isosugklinous¸n akolouji¸n, an → a.Ac upojèsoume, epiplèon, ìti o a = sup A den eÐnai stoiqeÐo tou A. Up�rqei

a1 ∈ A pou ikanopoieÐ thn a − 1 < a1 ≤ a. 'Omwc, a1 6= a (diìti a /∈ A), �raa− 1 < a1 < a.

Ac upojèsoume ìti èqoume breÐ a1, . . . , am ∈ A pou ikanopoioÔn ta ex c:

(i) a1 < a2 < · · · < am < a.

(ii) Gia k�je k = 1, . . . , m isqÔei a− 1k < ak < a.

Tìte, o sm = max{a− 1m+1 , am} eÐnai mikrìteroc apì ton a. MporoÔme loipìn

na broÔme am+1 ∈ A pou ikanopoieÐ thn sm < am+1 < a (ejhg;hste giatÐ). 'Ara,am < am+1 kai a− 1

m+1 < am+1 < a.Epagwgik�, orÐzetai gnhsÐwc aÔxousa akoloujÐa (an) stoiqeÐwn tou A pou

ikanopoioÔn thn a− 1n < an < a. Apì to krit rio twn isosugklinous¸n akolou-

ji¸n, an → a.

13. 'Estw x ∈ R. Up�rqei q1 ∈ Q pou ikanopoieÐ thn x − 1 < q1 < x (apì thnpuknìthta tou Q sto R).

Ac upojèsoume ìti èqoume breÐ rhtoÔc arijmoÔc q1, . . . , qm pou ikanopoioÔnta ex c:

(i) q1 < q2 < · · · < qm < x.

(ii) Gia k�je k = 1, . . . , m isqÔei x− 1k < qk < x.

Tìte, o sm = max{x − 1m+1 , qm} eÐnai mikrìteroc apì ton x. Lìgw thc puknì-

thtac twn rht¸n stouc pragmatikoÔc arijmoÔc, mporoÔme na broÔme qm+1 ∈ Qsto anoiktì di�sthma (sm, x). Tìte, qm < qm+1 kai x− 1

m+1 < qm+1 < x.Epagwgik�, orÐzetai gnhsÐwc aÔxousa akoloujÐa (qn) rht¸n arijm¸n pou ika-

nopoioÔn thn x− 1n < qn < x. Apì to krit rio twn isosugklinous¸n akolouji¸n,

qn → x.


(i) H (an) orÐzetai kal�. ArkeÐ na deÐxete ìti an ≥ 0 gia k�je n ∈ N. DeÐxteto me epagwg .


am+1 =√

2 +√

am ≤√

2 +√

am+1 = am+2.

(iii) H (an) eÐnai �nw fragmènh (me epagwg ). Parathr ste ìti an am ≤ 2 tìteam+1 =

√2 +

√am ≤

√2 +

√2 <

√2 + 2 = 2.

AfoÔ h (an) eÐnai aÔxousa kai �nw fragmènh, sugklÐnei.

15. Parathr ste pr¸ta ìti, an h (an) sugklÐnei tìte to ìrio thc ja ikanopoieÐthn x = 2x+1

x+1 (exhg ste giatÐ). 'Ara x = 1+√

52 x = 1−√5

2 . DeÐxte diadoqik�ta ex c:

(i) H (an) orÐzetai kal�. ArkeÐ na deÐxete ìti an 6= −1 gia k�je n ∈ N. DeÐxteme epagwg ìti an > 0 gia k�je n ∈ N.


am+2 − am+1 =am+1 − am

(am+1 + 1)(am + 1)≥ 0.

(iii) H (an) eÐnai �nw fragmènh (me epagwg ). Parathr ste ìti

am+1 =2am + 1am + 1

≤ 2am + 2am + 1

= 2

gia k�je m. Ja mporoÔsate epÐshc na deÐxete ìti am ≤ 1+√

52 gia k�je

m (afoÔ, apì ta parap�nw, autì eÐnai to {upoy fio ìrio} thc aÔxousacakoloujÐac (an)).

AfoÔ h (an) eÐnai aÔxousa kai �nw fragmènh, sugklÐnei (ston 1+√

52 ).


(i) H (xn) orÐzetai kal�. ArkeÐ na deÐxete ìti xn 6= 0 gia k�je n ∈ N. DeÐxteme epagwg ìti xn > 0 gia k�je n.

(ii) Gia k�je n ≥ 2 isqÔei xn ≥√

a (me epagwg ). Parathr ste ìti

xn+1 =12

(xn +

a

xn

)≥

√xn · a

xn=√

a

gia k�je n ∈ N.(iii) Gia k�je n ≥ 2 isqÔei xn ≥ xn+1 (me epagwg ). Parathr ste ìti

xn − xn+1 =x2

n − a

2xn≥ 0

gia k�je n ∈ N.AfoÔ h (xn)n≥2 eÐnai fjÐnousa kai k�tw fragmènh, sugklÐnei. To ìrio x eÐnaijetikì (apì ta prohgoÔmena èqoume x ≥ √

a) kai prèpei na ikanopoieÐ thn x =12

(x + a

x

), dhlad x2 = a. 'Ara, x =

√a.

17. K�noume pr¸ta thn epiplèon upìjesh ìti a = 0 kai deÐqnoume ìti bn → 0.JewroÔme ε > 0 kai brÐskoume n1(ε) ∈ N me thn idiìthta: gia k�je n ≥ n1 isqÔei|an| < ε/2. Tìte, gia k�je n > n1 èqoume

|bn| ≤ |a1 + · · ·+ an1 |n

+n− n1

n

ε

2<|a1 + · · ·+ an1 |

n+

ε

2.

O arijmìc A := |a1 + · · ·+ an1 | exart�tai apì to ε (afoÔ o n1 exart�tai apì toε) ìqi ìmwc apì to n. Apì thn Arqim deia idiìthta, up�rqei n2(A) = n2(ε) ∈ Nme thn idiìthta: gia k�je n ≥ n2 èqoume

|a1 + · · ·+ an1 |n

=A

n<

ε

2.

An loipìn p�roume n0 = max{n1, n2} tìte, gia k�je n ≥ n0 isqÔei h

|bn| ≤ A

n+

ε

2< ε.

Me b�sh ton orismì, bn → 0.

Gia th genik perÐptwsh, jewr ste thn akoloujÐa a′n := an − a. Tìte,a′n → 0. 'Ara,

bn − a =a1 + · · ·+ an

n− a =

(a1 − a) + · · ·+ (an − a)n

=a′1 + · · ·+ a′n

n→ 0.

'Epetai ìti bn → a.

18. Ac upojèsoume ìti èqoume breÐ fusikoÔc k1 < · · · < km pou ikanopoioÔn taex c:

0 < akm< akm−1 < · · · < ak1 kai 0 < akm

<1m

.

Jètoume εm+1 = min{a1, . . . , akm

, 1m+1

}> 0. AfoÔ inf(A) = inf{an : n ∈

N} = 0 kai εm+1 > 0, up�rqei km+1 ∈ N tètoioc ¸ste akm+1 < εm+1. Apì tonorismì tou εm+1 èpetai ìti (exhg ste giatÐ):

(a) km < km+1, (b) akm+1 < akm , (g) akm+1 <1

m + 1.

'Etsi orÐzetai epagwgik� mia gnhsÐwc fjÐnousa upakoloujÐa (akn) thc (an) hopoÐa sugklÐnei sto 0.

19. Upojètoume ìti oi upakoloujÐec (a2k) kai (a2k−1) sugklÐnoun ston a.'Estw ε > 0. Up�rqei n1 ∈ N me thn idiìthta: gia k�je k ≥ n1 isqÔei

|a2k − a| < ε. EpÐshc, up�rqei n2 ∈ N me thn idiìthta: gia k�je k ≥ n2 isqÔei|a2k−1 − a| < ε. An jèsoume n0 = max{2n1, 2n2 − 1} tìte gia k�je n ≥ n0

isqÔei |an − a| < ε.[Pr�gmati, parathr ste ìti an o n eÐnai �rtioc tìte n = 2k gia k�poion k ≥ n1

en¸ an o n eÐnai perittìc tìte n = 2k − 1 gia k�poion k ≥ n2.]AfoÔ to ε > 0 tan tuqìn, èpetai ìti an → a.

To antÐstrofo eÐnai aplì: èqoume dei ìti an mia akoloujÐa (an) sugklÐneiston a ∈ R tìte k�je upakoloujÐa (akn) thc (an) sugklÐnei ston a.

20. Ac upojèsoume ìti a2k → x, a2k−1 → y kai a3k → z. Parathr ste ìti:(i) H (a6k) eÐnai tautìqrona upakoloujÐa thc (a2k) kai upakoloujÐa thc (a3k).'Ara, h (a6k) sugklÐnei kai x = lim

k→∞a2k = lim

k→∞a6k = lim

k→∞a3k = z.

(ii) H (a6k−3) eÐnai tautìqrona upakoloujÐa thc (a2k−1) kai upakoloujÐa thc(a3k). 'Ara, h (a6k−3) sugklÐnei kai y = lim



k→∞a3k =

z.'Epetai ìti x = y = z. AfoÔ oi (a2k) kai (a2k−1) èqoun to Ðdio ìrio, h 'Askhsh19 deÐqnei ìti h (an) sugklÐnei.

21. DeÐxte diadoqik� ìti:

(a) H (a2n) eÐnai aÔxousa kai �nw fragmènh apì ton a1 en¸ h (a2n−1) eÐnaifjÐnousa kai k�tw fragmènh apì ton a2.

(b) Up�rqoun a, b ∈ R tètoioi ¸ste a2n → a kai a2n−1 → b.

(g) a = b.

(d) an → a = b.

(e) a2n ≤ a ≤ a2n−1 gia k�je n ∈ N.

23. DeÐxte tic isodunamÐec (1) ⇐⇒ (2) ⇐⇒ (3) ⇐⇒ (4):

(1) H akoloujÐa (an) den sugklÐnei ston a.

(2) Up�rqei ε > 0 ¸ste �peiroi ìroi am thc (an) na ikanopoioÔn thn |am−a| ≥ε.

(3) Up�rqoun ε > 0 kai fusikoÐ arijmoÐ k1 < k2 < · · · < kn < · · · ¸ste: giak�je n ∈ N isqÔei |akn

− a| ≥ ε.

(4) Up�rqoun ε > 0 kai upakoloujÐa (akn) thc (an) ¸ste: gia k�je n ∈ NisqÔei |akn − a| ≥ ε.

24. (=⇒) Upojèste pr¸ta ìti an → a. An (akn) eÐnai mia upakoloujÐa thc (an)tìte akn → a. 'Ara, ìlec oi upakoloujÐec thc (akn) sugklÐnoun ki autèc ston a.

(⇐=) Me apagwg se �topo: upojèste ìti h (an) den sugklÐnei ston a. Apìthn 'Askhsh 23, up�rqoun ε > 0 kai upakoloujÐa (akn) thc (an) ¸ste: gia k�jen ∈ N isqÔei |akn − a| ≥ ε. Tìte, an p�roume opoiad pote upakoloujÐa thc(akn), ìloi oi ìroi thc ja eÐnai ε-makri� apì ton a. Dhlad , h (akn) den èqeiupakoloujÐa pou na sugklÐnei ston a.

25. Parathr ste ìti: gia k�je n ∈ N,1

n + 1+

1n + 2

+ · · ·+ 12n

≥ 12n

+ · · ·+ 12n

=n

2n=

12.

Ac upojèsoume ìti h akoloujÐa an = 1 + 12 + · · · + 1

n eÐnai akoloujÐa Cauchy.Tìte, paÐrnontac ε = 1

4 > 0, mporoÔme na broÔme n0 ∈ N me thn idiìthta: giak�je m, n ≥ n0 isqÔei |am − an| < 1

4 .P�rte n ≥ n0 kai m = 2n ≥ n0. Tìte, |a2n − an| < 1

4 . Autì den mporeÐ naisqÔei, diìti

|a2n − an| =(

1 +12

+ · · ·+ 1n

+1

n + 1+ · · ·+ 1

2n

)−

(1 +

12

+ · · ·+ 1n

)

=1

n + 1+

1n + 2

+ · · ·+ 12n

≥ 12.

Exhg ste t¸ra ta ex c:

(a) AfoÔ h (an) den eÐnai akoloujÐa Cauchy, h (an) den sugklÐnei se pragma-tikì arijmì.

(b) AfoÔ h (an) eÐnai aÔxousa kai den sugklÐnei, h (an) den eÐnai �nw fragmènh.

(g) AfoÔ h (an) eÐnai aÔxousa kai den eÐnai �nw fragmènh, anagkastik� an →+∞.

26. Apì thn |an+1 − an| ≤ µ|an − an−1| èpetai (exhg ste giatÐ) ìti: gia k�jen ≥ 2 isqÔei

|an+1 − an| ≤ µn−1|a2 − a1| = |b− a| · µn−1.

An loipìn m,n ∈ N kai m > n, tìte

|am − an| ≤ |am − am−1|+ · · ·+ |an+1 − an|≤ |b− a| (µn−1 + · · ·+ µm−2

)

= |b− a| · µn−1 · 1− µm−n

1− µ

≤ |b− a|µn−1

1− µ

=|b− a|

µ(1− µ)· µn.

Jewr ste ε > 0 kai breÐte n0 ∈ N pou ikanopoieÐ thn |b−a|µ(1−µ)µ

n0 < ε. Tètoiocn0 up�rqei giatÐ µn → 0 ìtan n →∞. Tìte, an m > n ≥ n0,

|am − an| ≤ |b− a|µ(1− µ)

· µn ≤ |b− a|µ(1− µ)

· µn0 < ε.

Dhlad , h (an) eÐnai akoloujÐa Cauchy.

27. Parathr ste ìti: gia k�je n ≥ 2 isqÔei

an+1 − an =an + an−1

2− an = −an − an−1

2,

dhlad

|an+1 − an| ≤ 12|an − an−1|.

Apì thn 'Askhsh 26, h (an) eÐnai akoloujÐa Cauchy.

28. 'Estw ε > 0 kai èstw n0 ∈ N. DeÐxte ìti ta ex c eÐnai isodÔnama:

(a) Gia k�je m,n ≥ n0 isqÔei |am − an| ≤ ε.

(b) Gia k�je m > n ≥ n0 isqÔei |am − an| ≤ ε.

(g) Gia k�je n ≥ n0 kai k�je k ∈ N isqÔei |an+k − an| ≤ ε.

(d) Gia k�je n ≥ n0 isqÔei bn := sup{|an+k − an| : k ∈ N} ≤ ε.

Qrhsimopoi¸ntac thn isodunamÐa twn (a) kai (d) deÐxte ìti h (an) eÐnai akoloujÐaCauchy (isodÔnama, sugklÐnei) an kai mìno an bn → 0.

G. Ask seic

1. (a) OrÐzoume a = max{a1, a2, . . . , ak}. Tìte, gia k�je n ∈ N èqoume an <

an1 + · · ·+ an

k ≤ kan. 'Ara,

a < bn := n√

an1 + an

2 + · · ·+ ank ≤

n√

kan = an√

k.

AfoÔ limn→∞

n√

k = 1 (blèpe basik �skhsh 9), apì to krit rio twn isosugklinou-s¸n akolouji¸n èqoume bn → a. Gia par�deigma,

n√

2n + 3n + 7n → 7.

(b) To pl joc twn prosjetèwn (ston orismì tou n-ostoÔ ìrou) den eÐnai staje-rì. Doulèyte ìmwc ìpwc sto (a): parathr ste ìti nn < 1n+2n+· · ·+nn < n·nn

an n ≥ 2. 'Ara,

1 < xn =1n

n√

1n + 2n + · · ·+ nn < n√

n

gia k�je n ≥ 2. AfoÔ n√

n → 1, efarmìzetai to krit rio twn isosugklinous¸nakolouji¸n, kai xn → 1.

2. Qrhsimopoi ste to gegonìc ìti xn =(1 + 1

n

)n → e. Gia par�deigma,

(a) an =(1 + 1

n−1

)n−1

= xn−1 → e.

(b) bn =(1 + 2

n

)n =(

n+2n+1

)n (n+1

n

)n = n+1n+2xn+1xn → e2.

(g) 1cn

=(

nn−1

)n

= nn−1xn−1 → e, �ra cn → 1

e .

(d) dn =(1− 1

n2

)n =(1− 1

n

)n (1 + 1

n

)n → 1e · e = 1.

(e) e3n =

(1 + 2

3n

)3n → e2 (giatÐ?), �ra en → 3√

e2.


(i) an > 0 kai bn > 0 gia k�je n ∈ N.

(ii) an ≤ bn gia k�je n ∈ N. Apì ton anadromikì orismì (anex�rthta m�listaapì to poioÐ eÐnai oi an kai bn) èqete

an+1 =√

anbn ≤ an + bn

2= bn+1.

(iii) H (an) eÐnai aÔxousa. Parathr ste ìti an+1 =√

anbn ≥√

a2n = an gia

k�je n ∈ N.

(iv) H (bn) eÐnai fjÐnousa. Parathr ste ìti bn+1 = an+bn

2 ≤ 2bn

2 = bn giak�je n ∈ N.

Apì ta parap�nw, h (an) eÐnai aÔxousa kai �nw fragmènh apì ton b1, en¸ h (bn)eÐnai fjÐnousa kai k�tw fragmènh apì ton a1 (exhg ste giatÐ). 'Ara, up�rqouna, b ∈ R ¸ste an → a kai bn → b. Apì thn bn+1 = an+bn

2 èpetai ìti b = a+b2 ,

dhlad a = b.

4. Parathr ste ìti

xn+2 − xn+1 = −xn+1 − xn

3.

'Ara, h akoloujÐa yn = xn+1−xn ikanopoieÐ thn anadromik sqèsh yn+1 = −yn

3 .'Epetai (exhg ste giatÐ) ìti

yn =(−1

3

)n−1

y1 =(−1

3

)n−1

(b− a).

Parathr ste ìti

xn = x1 + (x2 − x1) + · · ·+ (xn − xn−1) = a + y1 + · · ·+ yn−1.

'Ara,

xn = a +

(1 +

(−1

3

)+

(−1

3

)2

+ · · ·+(−1

3

)n−2)

(b− a)

= a +1− (− 1

3 )n−1

1− (− 13 )

(b− a) → a +34(b− a) =

3b + a

4.

5. Upojètoume ìti limn→∞an+1−an

bn+1−bn= λ ìpou λ ∈ R (h perÐptwsh λ = +∞

exet�zetai an�loga). 'Estw ε > 0. Qrhsimopoi¸ntac kai to gegonìc ìti bn →+∞, blèpoume ìti up�rqei n1(ε) ∈ N ¸ste: gia k�je n ≥ n1 isqÔei bn > 0 kai

λ− ε

2<

an+1 − an

bn+1 − bn< λ +

ε

2.

AfoÔ h (bn) eÐnai gnhsÐwc aÔxousa, èqoume bn+1−bn > 0. 'Ara, gia k�je n ≥ n1

isqÔei (λ− ε

2

)(bn+1 − bn) < an+1 − an <

(λ +

ε

2

)(bn+1 − bn).

'Epetai (exhg ste giatÐ) ìti: gia k�je n > n1 isqÔei(λ− ε

2

)(bn − bn1) < an − an1 <

(λ +

ε

2

)(bn − bn1).

Diair¸ntac me bn paÐrnoume

(λ− ε

2

) (1− bn1

bn

)+

an1

bn<

an

bn<

(λ +

ε

2

) (1− bn1

bn

)+

an1

bn.

Parathr ste ìti

limn→∞

[(λ− ε

2

) (1− bn1

bn

)+

an1

bn

]= λ− ε

2

kai

limn→∞

[(λ +

ε

2

) (1− bn1

bn

)+

an1

bn

]= λ +

ε

2

giatÐ bn → +∞ ìtan n → ∞. 'Ara (exhg ste giatÐ) up�rqei n2 ∈ N, pouexart�tai apì to n1 kai apì to ε, ¸ste: gia k�je n ≥ n2 isqÔei

λ− ε <(λ− ε

2

) (1− bn1

bn

)+

an1

bn<

(λ +

ε

2

) (1− bn1

bn

)+

an1

bn< λ + ε.

Sunep¸c, an n ≥ n0 := max{n1, n2}, èqoume

λ− ε <an

bn< λ + ε.

AfoÔ to ε > 0 tan tuqìn, sumperaÐnoume ìti

limn→∞

an

bn= λ.

6. AfoÔ 1an→ 1

a , h basik �skhsh 17 deÐqnei ìti

1bn

=1a1

+ · · ·+ 1an

n→ 1

a.

Ara, bn → a. Gia thn γn, parathr ste ìti bn ≤ γn ≤ δn := a1+···+an

n apì thnanisìthta armonikoÔ-gewmetrikoÔ-arijmhtikoÔ mèsou, kai efarmìste to krit riotwn isosugklinous¸n akolouji¸n se sunduasmì me thn 'Askhsh 17.

7. Parathr ste ìti

an

n=

(an − an−1) + · · ·+ (a2 − a1) + a1

n=

bn−1 + · · ·+ b1

n+

a1

n

ìpou bn := an+1 − an → a. T¸ra, qrhsimopoi ste thn 'Askhsh 17.

8. ArkeÐ na deÐxoume ìti h (an) eÐnai �nw fragmènh. Tìte, h (an) sugklÐnei kai,apì thn 'Askhsh 17, lim

n→∞an = lim

n→∞bn = a.

K�noume pr¸ta thn epiplèon upìjesh ìti a1 ≥ 0. Tìte, an ≥ a1 ≥ 0 giak�je n ∈ N. AfoÔ h (bn) sugklÐnei, eÐnai fragmènh: eidikìtera, up�rqei M ∈ R¸ste: gia k�je k ∈ N isqÔei kbk = a1 + · · ·+ ak ≤ kM . PaÐrnontac k = 2n kaiqrhsimopoi¸ntac thn upìjesh ìti h (an) eÐnai aÔxousa, gr�foume

nan ≤ an+1 + · · ·+ a2n ≤ a1 + · · ·+ a2n = 2nb2n ≤ 2nM.

Dhlad , h (an) eÐnai �nw fragmènh apì ton 2M . PoÔ qrhsimopoi jhke h upìjeshìti oi ìroi thc (an) eÐnai mh arnhtikoÐ?

Gia th genik perÐptwsh, jewr ste thn (aÔxousa akoloujÐa) a′n = an − a1

kai thn b′n := a′1+···+a′nn . Apì thn upìjesh èqoume b′n → a−a1 kai, ìpwc orÐsthke

h (a′n), èqoume a′1 = 0. 'Ara, h (a′n) eÐnai �nw fragmènh. 'Epetai ìti h (an) eÐnai�nw fragmènh (exhg ste giatÐ).

9. DeÐxte pr¸ta ìti h (an) orÐzetai kal� kai an > 0 gia k�je n ∈ N. EpÐshc,deÐxte ìti an an → x tìte x =

√3.

(i) Upojèste ìti 0 < a1 <√

(a) a2 >√

3.

(b) Gia k�je n ∈ N isqÔei h an+2 = 3+2an

2+an.

(g) Gia k�je k ∈ N isqÔoun oi a2k−1 <√

3 kai a2k−1 < a2k+1.

(d) Gia k�je k ∈ N isqÔoun oi a2k >√

3 kai a2k+2 < a2k.

Qrhsimopoi¸ntac thn 'Askhsh 21 deÐxte ìti oi (a2k−1) kai (a2k) sugklÐnoun.Qrhsimopoi¸ntac thn anadromik sqèsh an�mesa ston an+2 kai ton an deÐxte ìtilim

k→∞a2k = lim

k→∞a2k−1 =

√3. Apì thn 'Askhsh 19 èpetai ìti an →

√3.

(ii) Exet�ste me ton Ðdio trìpo thn perÐptwsh a1 >√

3.

(iii) Tèloc, deÐxte ìti an a1 =√

3 tìte èqoume an =√

3 gia k�je n ∈ N.

6.4 Sunart seic

1. Upojètoume ìti x, y ∈ [0, 1] kai f(x) = f(y). Tìte,

a + (b− a)x = a + (b− a)y =⇒ (b− a)x = (b− a)y =⇒ x = y.

'Ara, h f eÐnai 1-1.

Gia na deÐxoume ìti h f eÐnai epÐ, jewroÔme tuqìn z ∈ [a, b] kai zht�me x ∈ [0, 1]me thn idiìthta f(x) = z. IsodÔnama, a + (b − a)x = z. H monadik lÔsh aut cthc exÐswshc eÐnai o x = z−a

b−a , o opoÐoc an kei sto [0, 1]: afoÔ a ≤ z ≤ b, èqoume0 ≤ z − a ≤ b− a �ra 0 ≤ z−a

b−a ≤ 1.

2. Parathr ste pr¸ta ìti an x ∈ [0, 1] tìte f(x) = 1−x1+x ∈ [0, 1] kai an t ∈ [0, 1]

tìte g(t) = 4t(1− t) ∈ [0, 1]. Dhlad , oi f kai g orÐzontai kal�.

(a) 'Eqoume

(f ◦ g)(x) = f(g(x)) =1− g(x)1 + g(x)

=1− 4x(1− x)1 + 4x(1− x)

kai

(g ◦ f)(x) = g(f(x)) = 4f(x)(1− f(x)) = 41− x

1 + x

(1− 1− x

1 + x

)=

8x(1− x)(1 + x)2

.

(b) H g den eÐnai 1-1. Gia par�deigma, g(1/3) = g(2/3) = 8/9 kai genik� g(t) =g(1− t) gia k�je t ∈ [0, 1]. 'Ara, h g−1 den orÐzetai.

H f eÐnai 1-1: an x, y ∈ [0, 1] kai 1−x1+x = 1−y

1+y tìte (1−x)(1+y) = (1−y)(1+x)dhlad 1− x + y − xy = 1− y + x− xy. 'Epetai ìti x = y. AfoÔ h f eÐnai 1-1,orÐzetai h f−1 sto f([0, 1]). MporeÐte m�lista na deÐxete ìti h f eÐnai epÐ kai ìtih f−1 : [0, 1] → [0, 1] dÐnetai apì thn f−1(z) = 1−z

1+z . Me �lla lìgia, f−1 = f .

3. Upojètoume ìti oi g : X → Y kai f : Y → Z eÐnai 1-1 kai epÐ. Zht�meh : Z → X ¸ste h ◦ (f ◦ g) = IdX kai (f ◦ g) ◦ h = IdZ . H 'Askhsh mac èqei


dh d¸sei thn h. H g−1 ◦ f−1 : Z → X orÐzetai kal� giatÐ oi f kai g eÐnaiantistrèyimec kai f−1 : Z → Y , g−1 : Y → X. Tèloc,

(g−1◦f−1)◦(f◦g) = g−1◦(f−1◦f)◦g = g−1◦IdY ◦g = g−1◦(IdY ◦g) = g−1◦g = IdX

kai

(f◦g)◦(g−1◦f−1) = f◦(g◦g−1)◦f−1 = f◦IdY ◦f−1 = f◦(IdY ◦f−1) = f◦f−1 = IdZ .

ShmeÐwsh: Gia na deÐxete ìti h f ◦ g eÐnai 1-1 kai epÐ:(a) An x1, x2 ∈ X kai (f ◦ g)(x1) = (f ◦ g)(x2), tìte f(g(x1)) = f(g(x2)) �rag(x1) = g(x2) diìti h f eÐnai 1-1. 'Omoia, apì thn g(x1) = g(x2) blèpoume ìtix1 = x2, afoÔ h g eÐnai 1-1. Autì apodeiknÔei ìti h f ◦ g eÐnai 1-1.(b) An z ∈ Z tìte up�rqei y ∈ Y ¸ste f(y) = z (giatÐ h f eÐnai epÐ) kai up�rqeix ∈ X ¸ste g(x) = y (giatÐ h g eÐnai epÐ). Tìte, (f ◦g)(x) = f(g(x)) = f(y) = z.Autì apodeiknÔei ìti h f ◦ g eÐnai epÐ.

4. 'Estw g : X → Y kai f : Y → Z dÔo sunart seic.(a) 'Estw ìti h f ◦ g eÐnai epÐ. Gia na deÐxoume ìti h f eÐnai epÐ jewroÔme tuqìnz ∈ Z kai zht�me y ∈ Y ¸ste f(y) = z. AfoÔ h f ◦ g eÐnai epÐ, up�rqei x ∈ X

¸ste f(g(x)) = (f ◦ g)(x) = z. PaÐrnontac y = g(x) ∈ Y èqoume to zhtoÔmeno.To antÐstrofo tou (a) den isqÔei: dokim�ste tic f(x) = x kai g(x) = 1 (apì toR sto R).(b) 'Estw ìti h f ◦ g eÐnai 1-1. Gia na deÐxoume ìti h g eÐnai 1-1 jewroÔmex1, x2 ∈ X me g(x1) = g(x2) kai deÐqnoume ìti x1 = x2. AfoÔ g(x1) = g(x2)èqoume (f ◦ g)(x1) = f(g(x1)) = f(g(x2)) = (f ◦ g)(x2). AfoÔ h f ◦ g eÐnai 1-1,èpetai ìti x1 = x2. To antÐstrofo tou (b) den isqÔei: dokim�ste tic g(x) = x

kai f(x) = 1 (apì to R sto R).

5. Apì thn f ◦ g = IdX kai apì thn 'Askhsh 4(a), h f eÐnai epÐ. Apì thnh ◦ f = IdX kai apì thn 'Askhsh 4(b), h f eÐnai 1-1. 'Ara, h f−1 : X → X

orÐzetai kal� kai ikanopoieÐ tic f−1 ◦ f = f ◦ f−1 = IdX . T¸ra, parathr steìti

h = h ◦ (f ◦ f−1) = (h ◦ f) ◦ f−1 = IdX ◦ f−1 = f−1

kaig = (f−1 ◦ f) ◦ g = f−1 ◦ (f ◦ g) = f−1 ◦ IdX = f−1.

Dhlad , h = g = f−1.

8. Parathr ste ìti: an h f gr�fetai sth morf f = fa + fp ìpou h fa eÐnai�rtia kai h fp peritt tìte, b�zontac ìpou x ton −x sthn f(x) = fa(x) + fp(x),paÐrnoume

f(−x) = fa(−x) + fp(−x) = fa(x)− fp(x)

gia k�je x ∈ R. Dhlad , oi fa kai fp prèpei na ikanopoioÔn tic

f(x) = fa(x) + fp(x)

f(−x) = fa(x)− fp(x)

gia k�je x ∈ R. Anagkastik� (exhg ste giatÐ), oi fa kai fp prèpei na dÐnontaiapì tic

fa(x) =f(x) + f(−x)

2kai fp(x) =

f(x)− f(−x)2

.

Elègxte ìti an orÐsoume ètsi tic fa kai fp tìte h fa eÐnai �rtia, h fp eÐnai peritt kai f = fa +fp. H monadikìthta thc anapar�stashc èqei dh apodeiqjeÐ (giatÐ?).

9. (a) Ac upojèsoume ìti h f(x) = [x] eÐnai periodik me perÐodo a. MporoÔmena upojèsoume ìti a > 0 (an h f èqei perÐodo a tìte èqei kai perÐodo −a).Parathr ste ìti f(R) = f([0, a)). Pr�gmati, k�je x ∈ R gr�fetai sth morf x = mxa+yx gia k�poiouc mx ∈ Z kai yx ∈ [0, a), opìte, gia k�je x ∈ R èqoumef(x) = f(mxa + yx) = f(yx) ∈ f([0, a)).

'Omwc, f([0, a)) ⊆ [0, a): an 0 ≤ x < a tìte 0 ≤ [x] ≤ x < a. 'Ara, h f eÐnaifragmènh. Autì eÐnai �topo, afoÔ f(n) = n gia k�je n ∈ N.(b) H sun�rthsh f(x) = x − [x] eÐnai periodik me perÐodo 1. ArkeÐ na deÐxoumeìti [x+1] = [x]+1 gia k�je x ∈ R. 'Omwc, [x] ≤ x < [x]+1 �ra [x]+1 ≤ x+1 <

([x] + 1)+1. Dhlad , o akèraioc m = [x] + 1 ikanopoieÐ thn m ≤ x+1 < m+1.'Ara, [x] + 1 = m = [x + 1] (apì th monadikìthta tou akeraÐou mèrouc).

10. (a) Parathr ste ìti, apì thn 'Askhsh 9(b),

f(x +

1n

)=

[x +

1n

]+ · · ·+

[x +

1n

+n− 2

n

]+

[x +

1n

+n− 1

n

]−

[n(x +

1n

)]

=[x +

1n

]+ · · ·+

[x +

n− 1n

]+ [x + 1]− [nx + 1]

=[x +

1n

]+ · · ·+

[x +

n− 1n

]+ [x] + 1− [nx]− 1

= [x] +[x +

1n

]+ · · ·+

[x +

n− 1n

]− [nx]

= f(x).

(b) An 0 ≤ x < 1/n tìte

x, x +1n

, . . . , x +n− 1

n, nx ∈ [0, 1)

�ra

[x] =[x +

1n

]= · · · =

[x +

n− 1n

]= [nx] = 0.

'Epetai ìti f(x) = 0.

(g) K�je x ∈ R gr�fetai sth morf x = mx · 1n + yx gia k�poiouc mx ∈ Z

kai yx ∈ [0, 1/n), opìte, apì ta (a) kai (b), gia k�je x ∈ R èqoume f(x) =f(mx · 1

n + yx

)= f(yx) = 0.

11. 'Oqi. An orÐsete thn f : [0, 1] ∪ [2, 3] → R me f(x) = x an x ∈ [0, 1]kai f(x) = x − 3 an x ∈ [2, 3], tìte h f eÐnai gnhsÐwc aÔxousa sta diast mata

I1 = [0, 1] kai I2 = [2, 3]. 'Omwc, f(0) = 0 > −1 = f(2). 'Ara, h f den eÐnaignhsÐwc aÔxousa sto I1 ∪ I2.

12. H f orÐzetai kal� sto 1: 1 + 1 = 12 + 1. Parathr ste ìti

(a) An x < y ≤ 1 tìte f(x) = x + 1 < y + 1 = f(y).

(b) An 1 ≤ x < y tìte f(x) = x2 + 1 < y2 + 1 = f(y).

(g) An x < 1 < y tìte f(x) < f(1) < f(y) apì ta (a) kai (b).

Apì ta parap�nw blèpoume ìti an x, y ∈ R kai x < y tìte f(x) < f(y). 'Ara, hf eÐnai gnhsÐwc aÔxousa.

Gia to deÔtero er¸thma parathr ste ìti

(a) An y ≤ 2 tìte f(y − 1) = y.

(b) An y ≥ 2 tìte f(√

y − 1) = y.

Sunep¸c, h antÐstrofh sun�rthsh f−1 : R→ R orÐzetai apì tic f−1(y) = y − 1an y ≤ 2 kai f−1(y) =

√y − 1 an y ≥ 2.

13. H g prèpei na ikanopoieÐ thn g(f(x)) = f(g(x)) gia k�je x ∈ R. Dhlad ,thn

g(x + 1) = g(x) + 1.

Mia tètoia sun�rthsh eÐnai h g(x) = x. An h eÐnai opoiad pote periodik sun�r-thsh me perÐodo 1, tìte

(h + g)(x + 1) = h(x + 1) + g(x + 1) = h(x) + g(x) + 1 = (h + g)(x) + 1

gia k�je x ∈ R. 'Ara, h h + g ikanopoieÐ ki aut thn (h + g) ◦ f = f ◦ (h + g).

14. (a) ArkeÐ na deÐxoume ìti h f eÐnai gnhsÐwc aÔxousa sto (−∞, 0] kai sto[0, +∞). Exhg ste giatÐ (parìmoio epiqeÐrhma qrhsimopoi jhke sthn 'Askhsh12).Ac upojèsoume ìti x < y ≤ 0. Jèloume na elègxoume ìti x

1−x < y1−y . AfoÔ

1−x kai 1−y > 0, isodÔnama zht�me x(1−y) < y(1−x) dhlad x−xy < y−xy

(pou isqÔei).Me ton Ðdio trìpo deÐxte ìti an 0 ≤ x < y tìte x

x+1 < yy+1 .

(b) Parathr ste ìti

|f(x)| = |x||x|+ 1

< 1

gia k�je x ∈ R, dhlad f(R) ⊆ (−1, 1). DeÐxte ìti isqÔei isìthta: an 0 < y < 1tìte up�rqei x > 0 ¸ste f(x) = x

x+1 = y, en¸ an −1 < y < 0 tìte up�rqeix < 0 ¸ste f(x) = x

1−x = y.

15. 'Eqoume upojèsei ìti h f ikanopoieÐ thn f(x + y) = f(x) + f(y) gia k�jex, y ∈ R.

6.5 Suneqeia kai Oria · 179

(a) PaÐrnontac x = y = 0 mporeÐte na elègxete ìti f(0) = 0. Katìpin, giadosmèno x ∈ R, paÐrnontac y = −x mporeÐte na elègxete ìti f(−x) = −f(x).(b) Qrhsimopoi ste epagwg .(g) P�rte x1 = · · · = xn = 1

n sto (b).(d) Gr�yte ton q sth morf ±(

1n + · · ·+ 1

n

)� gia kat�llhlo pl joc prosjetèwn

� kai qrhsimopoi ste ta (b) kai (g).

16. ArkeÐ na deÐxoume ìti gia tuqìntec pragmatikoÔc arijmoÔc a kai b me a 0.

QwrÐzoume to [a, b] se n diadoqik� upodiast mata m kouc (b − a)/n me tashmeÐa a = x0 < x1 < · · · < xn−1 < xn = b, ìpou xk = a + k(b−a)

n , k =0, 1, . . . , n. ParathroÔme ìti, apì thn upìjesh,

|f(xk)− f(xk−1)| ≤ (xk − xk−1)2 =(

b− a

n

)2

=(b− a)2

n2

gia k�je k = 1, . . . , n. Apì thn trigwnik anisìthta gia thn apìluth tim èqoume

δ = |f(b)− f(a)| = |f(xn)− f(x0)|≤ |f(xn)− f(xn−1)|+ · · ·+ |f(x2)− f(x1)|+ |f(x1)− f(x0)|

≤ n · (b− a)2

n2=

(b− a)2

n.

Dhlad , δ ≤ (b − a)2/n gia k�je n ∈ N. Autì eÐnai �topo afoÔ δ > 0 kai(b−a)2

n → 0 ìtan to n →∞.

6.5 Sunèqeia kai 'Oria



1. Swstì. Efarmìzoume ton orismì thc sunèqeiac me ε = 15 > 0. AfoÔ h f eÐnai

suneq c sto x0, up�rqei δ > 0 ¸ste: gia k�je x ∈ (x0 − δ, x0 + δ) isqÔei

|f(x)− f(x0)| < 15, dhlad |f(x)− 1| < 1

5.

Sunep¸c, gia k�je x ∈ (x0 − δ, x0 + δ) isqÔei 45 = 1− 1

5 < f(x) < 1 + 15 = 6

5 .

2. Swstì. 'Ola ta shmeÐa tou pedÐou orismoÔ thc f eÐnai memonwmèna shmeÐatou, �ra h f eÐnai suneq c se aut�. To epiqeÐrhma eÐnai to ex c: èstw m ∈ N kaièstw ε > 0. Epilègoume δ = 1

2 . An n ∈ N kai |n −m| < 12 , tìte, anagkastik�,

n = m. Sunep¸c,

|f(n)− f(m)| = |f(m)− f(m)| = 0 < ε.

3. Swstì. An x0 /∈ N tìte up�rqei δ > 0 ¸ste sto (x0 − δ, x0 + δ) na mhnperièqetai fusikìc arijmìc (exhg ste giatÐ). Tìte, gia k�je ε > 0 kai gia k�jex ∈ (x0 − δ, x0 + δ) èqoume |f(x) − f(x0)| = |1 − 1| = 0 < ε. 'Ara, h f eÐnaisuneq c sto x0.

'Estw t¸ra x0 ∈ N. Up�rqei akoloujÐa (xn) h opoÐa sugklÐnei sto x0 kaih opoÐa den èqei ìrouc pou na eÐnai fusikoÐ arijmoÐ (exhg ste giatÐ). Tìte,f(xn) = 1 → 1 6= 0 = f(x0). SÔmfwna me thn arq thc metafor�c, h f den eÐnaisuneq c sto x0.

4. Swstì. Jewr ste th sun�rthsh f : R→ R pou paÐrnei thn tim 1 sta shmeÐa0, 1, 1

2 , . . . , 1n , . . . kai thn tim 0 se ìla ta �lla shmeÐa.

5. Swstì. Jewr ste th sun�rthsh f : R → R pou orÐzetai wc ex c: f(x) = x

an x = 1, 12 , . . . , 1

n , . . . kai f(x) = 0 se ìla ta �lla shmeÐa. Gia na deÐxete ìti hf eÐnai suneq c sto shmeÐo 0 qrhsimopoi ste ton ε− δ orismì thc sunèqeiac.

6. Swstì. Jewr ste th sun�rthsh f : R → R me f(x) = x an x ∈ Q kaif(x) = −x an x /∈ Q.

7. L�joc. Jewr ste th sun�rthsh f : R → R me f(x) = 0 an x 6= 4 kaif(4) = 1. H f eÐnai asuneq c sto x0 = 4 kai suneq c se k�je �rrhto x.

8. Swstì. Jewr ste x ∈ (a, b) kai akoloujÐa (qn) rht¸n arijm¸n apì to (a, b)h opoÐa sugklÐnei sto x. Tètoia akoloujÐa up�rqei lìgw thc puknìthtac twnrht¸n sto R. AfoÔ h f eÐnai suneq c sto x, h arq thc metafor�c deÐqnei ìtif(qn) → f(x). Apì thn upìjesh èqoume f(qn) = 0 gia k�je n ∈ N, kai sunep¸c,f(x) = 0.

9. L�joc. Jewr ste th sun�rthsh f : (0, 1] → R me f(x) = 1x . H f eÐnai

suneq c sto (0, 1]. An ìmwc jewr soume thn akoloujÐa xn = 1n , tìte h (xn)

eÐnai akoloujÐa Cauchy sto pedÐo orismoÔ thc f , en¸ h f(xn) = n den eÐnaiakoloujÐa Cauchy.

10. Swstì. 'Eqoume 12n → 0 kai 1

2n−1 → 0. Apì thn upìjesh, f(

12n

)=

(−1)2n = 1 → 1 kai f(

12n−1

)= (−1)2n−1 = −1 → −1. Apì thn arq thc

metafor�c, h f eÐnai asuneq c sto shmeÐo 0.

11. Swstì. An f(0) = 0 tìte paÐrnoume x0 = 0 1. An f(0) 6= 0, tìte apìthn f(0) = −f(1) blèpoume ìti h f paÐrnei eterìshmec timèc sta �kra tou [0, 1].Apì to je¸rhma endi�meshc tim c, up�rqei x0 ∈ (0, 1) ¸ste f(x0) = 0.

12. L�joc. Jewr ste thn f : (0, 1) → R me f(x) = x. H f eÐnai suneq c sto(0, 1), ìmwc den paÐrnei mègisth oÔte el�qisth tim sto (0, 1).

13. Swstì. 'Ena apì ta basik� jewr mata gia suneqeÐc sunart seic pou èqounpedÐo orismoÔ kleistì di�sthma.

14. Swstì. Qrhsimopoi ste, gia par�deigma, thn arq thc metafor�c. Anxn 6= 0 kai xn → 0, tìte

∣∣∣∣ g(xn) sin1xn

∣∣∣∣ ≤ |g(xn)|.

AfoÔ limx→0

g(x) = 0, èqoume g(xn) → 0. Apì thn prohgoÔmenh anisìthta èpetaiìti

limx→0

g(xn) sin1xn

= 0.

15. Swstì. Me apagwg se �topo: an den isqÔei h limx→a+

f(x) = L, tìte

up�rqei ε > 0 me thn ex c idiìthta: gia k�je δ > 0 up�rqei x ∈ (a,+∞) mea < x < a + δ kai |f(x) − L| ≥ ε. Efarmìzontac to parap�nw me δ = 1brÐskoume x1 ∈ (a,+∞) me a < x1 < a + 1 kai |f(x1) − L| ≥ ε. Efarmìzontacto parap�nw me δ = min{1/2, x1 − a} brÐskoume x2 < x1 me a < x2 < a + 1

2

kai |f(x2)−L| ≥ ε. SuneqÐzontac me ton Ðdio trìpo, orÐzoume gnhsÐwc fjÐnousaakoloujÐa tn = xn − a me tn → 0 kai |f(a + tn)− L| ≥ ε. Autì eÐnai �topo.

16. L�joc. Jewr ste th sun�rthsh f : (0, +∞) → R pou paÐrnei thn tim 1 stashmeÐa 1, 1

2 , . . . , 1n , . . . kai thn tim 0 se ìla ta �lla shmeÐa. To lim

n→∞f(1/n) = 1,

ìmwc to limx→0+

f(x) den up�rqei.


1. (a) Parathr ste ìti f(x) = x2+x−6x−2 = x + 3 an x 6= 2. An (xn) eÐnai mia

akoloujÐa jetik¸n arijm¸n me xn → 2, tìte f(xn) = xn + 3 → 5 6= 0 = f(0).'Ara, h f den eÐnai suneq c sto shmeÐo x0 = 0. H f eÐnai suneq c se k�je x0 6= 0(grammik sun�rthsh).(b) GnwrÐzoume ìti sin x ≤ x ≤ tanx, �ra cos x ≤ sin x

x ≤ 1 gia x ∈ (0, π/2). An(xn) eÐnai mia akoloujÐa jetik¸n arijm¸n me xn → 0, tìte cosxn → cos 0 = 1.Apì to krit rio twn isosugklinous¸n akolouji¸n, f(xn) = sin xn

xn→ 1 6= f(0).

'Ara, h f den eÐnai suneq c sto shmeÐo x0 = 0. H f eÐnai suneq c se k�je x0 6= 0.(g) H f eÐnai asuneq c sto 0 an k ≤ 0: dokim�ste thn akoloujÐa xn = − 1

2πn+ π2.

H f eÐnai suneq c sto 0 an k ≥ 1: parathr ste ìti |f(x)| ≤ |x|k ≤ |x| gia k�jex ∈ [−1, 0].(d) H f den eÐnai suneq c sto shmeÐo 0: gia thn akoloujÐa xn = 1√

2πn+ π2

èqoume

xn → 0 kai f(xn) =√

2πn + π2 → +∞. H f eÐnai suneq c se k�je x0 6= 0.

(e) 'Estw x0 > 0 kai èstw ε > 0. Parathr ste ìti

|f(x)− f(x0)| = |√x−√x0| = |x− x0|√x +

√x0

≤ |x− x0|√x0

gia k�je x ∈ R+. An loipìn epilèxoume δ = ε√

x0 > 0 èqoume: gia k�je x ∈ R+

me |x− x0| < δ isqÔei |f(x)− f(x0)| < ε. Dhlad , h f eÐnai suneq c sto x0.

H f eÐnai suneq c kai sto shmeÐo 0. Autì prokÔptei eÔkola apì thn arq thcmetafor�c: an xn ∈ R+ kai xn → 0, tìte f(xn) =

√xn → 0 = f(0).

2. Ac upojèsoume ìti h f eÐnai suneq c sto x0 ∈ R. Up�rqei akoloujÐa(qn) rht¸n arijm¸n me qn → x0 kai up�rqei akoloujÐa (αn) arr twn arijm¸nme αn → x0. AfoÔ h f eÐnai suneq c sto x0 èqoume f(x0) = lim

n→∞f(qn) =

limn→∞

qn = x0 kai f(x0) = limn→∞

f(αn) = limn→∞

α3n = x3

0. 'Ara, x0 = x30. Autì

mporeÐ na sumbaÐnei mìno an x0 = −1, 0 1. Dhlad , h f eÐnai asuneq c se k�jex0 /∈ {−1, 0, 1}.

Ac upojèsoume ìti x0 ∈ {−1, 0, 1}. Tìte, an jewr soume tic suneqeÐc sunar-t seic g, h : R → R me g(x) = x kai h(x) = x3, èqoume f(x0) = g(x0) = h(x0).'Estw ε > 0. AfoÔ h g eÐnai suneq c sto x0, up�rqei δ1 > 0 ¸ste: an|x − x0| < δ1 tìte |g(x) − g(x0)| < ε. AfoÔ h h eÐnai suneq c sto x0, up�rqeiδ2 > 0 ¸ste: an |x−x0| < δ2 tìte |h(x)−h(x0)| < ε. Jètoume δ = min{δ1, δ2}.An |x− x0| < δ, tìte:

(i) eÐte x ∈ Q kai |x−x0| < δ ≤ δ1, opìte |f(x)−f(x0)| = |g(x)− g(x0)| < ε,

(ii) x /∈ Q kai |x− x0| < δ ≤ δ2, opìte |f(x)− f(x0)| = |h(x)− h(x0)| < ε.

Se k�je perÐptwsh, |x − x0| < δ ⇒ |f(x) − f(x0)| < ε, kai afoÔ to ε > 0 tantuqìn, h f eÐnai suneq c sto x0.

3. (a) Upojètoume pr¸ta ìti f(x0) > 0. AfoÔ h f eÐnai suneq c sto x0, anjewr soume ton ε = f(x0)

2 > 0 up�rqei δ > 0 ¸ste: an x ∈ X kai |x − x0| < δ

tìte

|f(x)− f(x0)| < f(x0)2

⇒ −f(x0)2

< f(x)− f(x0) ⇒ 0 <f(x0)

2< f(x).

(b) Upojètoume t¸ra ìti f(x0) < 0. AfoÔ h f eÐnai suneq c sto x0, an jew-r soume ton ε = − f(x0)

2 > 0 up�rqei δ > 0 ¸ste: an x ∈ X kai |x − x0| < δ

tìte

|f(x)− f(x0)| < −f(x0)2

⇒ f(x)− f(x0) < −f(x0)2

⇒ f(x) <f(x0)

2< 0.

4. Sthn 'Askhsh 15 tou FulladÐou gia tic Sunart seic eÐdame ìti f(q) = f(1)q =αq gia k�je q ∈ Q. 'Estw x ∈ R. JewroÔme akoloujÐa rht¸n arijm¸n qn → x.AfoÔ h f eÐnai suneq c, apì thn arq thc metafor�c paÐrnoume

f(x) = limn→∞

f(qn) = limn→∞

(αqn) = αx.

5. (a) 'Estw y ∈ R. MporoÔme na broÔme akoloujÐa (xn) rht¸n arijm¸n mexn → y. AfoÔ h f eÐnai suneq c, apì thn arq thc metafor�c èqoume 0 =f(xn) → f(y). 'Ara, f(y) = 0.

(b) Efarmìste to (a) gia thn suneq sun�rthsh h = f − g.

(g) 'Estw y ∈ R. MporoÔme na broÔme akoloujÐa (xn) rht¸n arijm¸n me xn → y.Apì thn upìjesh èqoume f(xn) ≤ g(xn) gia k�je n ∈ N. Apì thn arq thcmetafor�c paÐrnoume

f(y) = limn→∞

f(xn) ≤ limn→∞

g(xn) = g(y).

6. 'Estw x ∈ R. Gia k�je n ∈ N, o akèraioc mn = [2nx] ikanopoieÐ thnmn ≤ 2nx < mn + 1. 'Ara,

mn

2n≤ 2nx

2n= x <

mn + 12n

=mn

2n+

12n

.

Dhlad , x = limn→∞

mn

2n . Apì thn arq thc metafor�c,

f(x) = limn→∞

f(mn

2n

)= lim

n→∞0 = 0.

7. An M = 0 tìte h f eÐnai stajer (giatÐ?). MporoÔme loipìn na upojèsoumeìti M > 0.

'Estw x0 ∈ X kai èstw ε > 0. Epilègoume δ = ε/M > 0. An x ∈ X kai|x− x0| < δ, tìte |f(x)− f(x0)| ≤ M |x− x0| < Mδ = ε. To ε > 0 tan tuqìn,�ra h f eÐnai suneq c sto x0.

8. (a) Parathr ste ìti |f(0)| ≤ 0, dhlad f(0) = 0. 'Estw (xn) akoloujÐa stoR me xn → 0. Tìte, −xn ≤ f(xn) ≤ xn �ra f(xn) → 0 = f(0). Apì thn arq thc metafor�c h f eÐnai suneq c sto 0.

(b) H sun�rthsh f(x) ={

x , x /∈ Q−x , x ∈ Q,

eÐnai suneq c mìno sto shmeÐo 0 (mporeÐte na to deÐxete douleÔontac ìpwcsthn 'Askhsh 2) kai ikanopoieÐ thn |f(x)| ≤ |x| gia k�je x ∈ R.(g) Parathr ste ìti |f(0)| ≤ |g(0)| = 0, dhlad f(0) = 0. 'Estw (xn) akoloujÐasto R me xn → 0. AfoÔ h g eÐnai suneq c sto 0, èqoume g(xn) → 0. Apì thnupìjesh èqoume −g(xn) ≤ f(xn) ≤ g(xn), �ra f(xn) → 0 = f(0). Apì thn arq thc metafor�c h f eÐnai suneq c sto 0.

9. Pr¸toc trìpoc. JewroÔme tuqìn x1 ∈ [a, b]. Apì thn upìjesh, up�rqeix2 ∈ [a, b] ¸ste |f(x2)| ≤ 1

2 |f(x1)|. Epagwgik�, mporoÔme na orÐsoume xn ∈ [a, b]¸ste

|f(xn)| ≤ 12|f(xn−1)| ≤ 1

22|f(xn−2)| ≤ · · · ≤ 1

2n−1|f(x1)|.

'Epetai ìti f(xn) → 0. T¸ra, qrhsimopoi ste to je¸rhma Bolzano-Weierstrass:h (xn) eÐnai fragmènh (èqoume a ≤ xn ≤ b) �ra èqei sugklÐnousa upakoloujÐa.Dhlad , up�rqei upakoloujÐa (xkn) thc (xn) h opoÐa sugklÐnei se k�poio x0 (kaia ≤ x0 ≤ b � exhg ste giatÐ). Apì thn arq thc metafor�c èqoume f(xkn) →f(x0). 'Omwc, h (f(xkn)) eÐnai upakoloujÐa thc (f(xn)) kai f(xn) → 0. 'Ara,f(xkn) → 0. To ìrio thc (f(xkn)) eÐnai monadikì, sunep¸c f(x0) = 0.

DeÔteroc trìpoc. Upojètoume ìti h f den mhdenÐzetai sto [a, b]. Tìte, |f(t)| > 0gia k�je t ∈ [a, b]. H suneq c sun�rthsh |f | paÐrnei el�qisth tim sto [a, b].Dhlad , up�rqei x ∈ [a, b] ¸ste

|f(t)| ≥ |f(x)| > 0 gia k�je t ∈ [a, b].

'Omwc, apì thn upìjesh, up�rqei y ∈ [a, b] ¸ste

0 < |f(y)| ≤ 12|f(x)| ≤ 1

2|f(y)|.

Dhlad , |f(y)| < 0, to opoÐo eÐnai �topo.

10. 'Estw x > 0. Gr�foume P (x) = amxm + · · ·+ a1x + a0 = amxm(1 + ∆(x))ìpou

∆(x) =am−1x

m−1 + · · ·+ a1x + a0

amxm.

Parathr ste ìti an

x > 2|am−1|+ · · ·+ |a1|+ |a0|

|am| + 1

tìte

|∆(x)| ≤ |am−1|xm−1 + · · ·+ |a1|x + |a0||am|xm

<|am−1|+ · · ·+ |a1|+ |a0|

|am|x <12.

'Ara, up�rqei M > 0 ¸ste oi P (M) kai am na èqoun to Ðdio prìshmo (exhg stegiatÐ). Qrhsimopoi¸ntac kai thn P (0) = a0, blèpoume ìti o P (M)P (0) eÐnaiomìshmoc me ton a0am, dhlad arnhtikìc. Apì to je¸rhma endi�meshc tim cup�rqei ρ ∈ (0,M) ¸ste P (ρ) = 0.

11. H f eÐnai suneq c sto [0, 1], �ra paÐrnei el�qisth tim m kai mègisth tim M sto [0, 1]. An upojèsoume ìti h f den eÐnai stajer sun�rthsh, tìte m < M

(giatÐ?). GnwrÐzoume ìti up�rqei �rrhtoc α ¸ste m < α < M . Apì to je¸rhmaendi�meshc tim c up�rqei x ∈ (0, 1) me f(x) = α. Autì èrqetai se antÐfash methn upìjesh ìti h f paÐrnei mìno rhtèc timèc.

12. Parathr ste ìti afoÔ h f den mhdenÐzetai se kanèna x ∈ [a, b] to Ðdio isqÔeikai gia thn g.

QwrÐc periorismì thc genikìthtac mporoÔme na upojèsoume ìti f(a) > 0.Tìte èqoume f(x) > 0 gia k�je x ∈ [a, b] (an h f èpairne k�pou arnhtik ti-m tìte, apì to je¸rhma endi�meshc tim c, ja up rqe kai shmeÐo sto opoÐo jamhdenizìtan).

Ac upojèsoume ìti g(a) > 0. 'Opwc prin, èqoume g(x) > 0 gia k�je x ∈ [a, b].AfoÔ f2(x) = g2(x) gia k�je x, sumperaÐnoume ìti g(x) = f(x) gia k�je x.Dhlad , g = f .

Elègxte thn perÐptwsh f(a) > 0 kai g(a) < 0 me ton Ðdio trìpo.

13. OrÐzoume g : [0, 1] → R me g(x) = f(x)− f(x + 1). H g eÐnai kal� orismènhkai suneq c sto [0, 1]. Parathr ste ìti

g(0) = f(0)− f(1) = f(2)− f(1) = −g(1),

�ra g(0)g(1) ≤ 0. Apì to je¸rhma endi�meshc tim c up�rqei x ∈ [0, 1] ¸ste

f(x)− f(x + 1) = g(x) = 0.

Jètontac y = x + 1 èqoume to zhtoÔmeno.

14. (a) H f eÐnai suneq c sto [a, b], �ra paÐrnei el�qisth tim m kai mègisthtim M sto [a, b]. Gia i = 1, 2 èqoume m ≤ f(xi) ≤ M , �ra

m = tm + (1− t)m ≤ tf(x1) + (1− t)f(x2) ≤ tM + (1− t)M = M.

Apì to je¸rhma endi�meshc tim c up�rqei yt ∈ [a, b] me f(yt) = tf(x1) + (1 −t)f(x2).

(b) H f eÐnai suneq c sto [a, b], �ra paÐrnei el�qisth tim m kai mègisth tim M

sto [a, b]. Gia k�je i = 1, . . . , n èqoume m ≤ f(xi) ≤ M , �ra

m ≤ α :=f(x1) + · · ·+ f(xn)

n≤ M.

Apì to je¸rhma endi�meshc tim c up�rqei y ∈ [a, b] me f(y) = α.

15. Jewr ste th suneq sun�rthsh g : [a, b] → R pou orÐzetai apì thn g(x) =f(x) − x. Parathr ste ìti g(a) = f(a) − a ≥ 0 kai g(b) = f(b) − b ≤ 0.AfoÔ g(a)g(b) ≤ 0, apì to je¸rhma endi�meshc tim c up�rqei x ∈ [a, b] ¸stef(x)− x = g(x) = 0.

16. Jewr ste th suneq sun�rthsh g : R → R pou orÐzetai apì thn g(x) =f(x)−x. Parathr ste ìti h g eÐnai gnhsÐwc fjÐnousa: an x < y tìte f(x) ≥ f(y)kai −x > −y, sunep¸c g(x) = f(x)−x > f(y)−y = g(y). Eidikìtera, h g eÐnai 1-1. An loipìn mhdenÐzetai se k�poio shmeÐo, tìte autì to shmeÐo ja eÐnai monadikì.Autì apodeiknÔei ìti up�rqei to polÔ mÐa lÔsh thc exÐswshc f(x) = x.

Parathr ste ìti: an x > 0 tìte f(x) ≤ f(0), �ra

g(x) = f(x)− x ≤ f(0)− x.

An loipìn p�roume x1 = |f(0)| + 1 tìte g(x1) < 0. 'Omoia, an x < 0 tìtef(x) ≥ f(0), �ra

g(x) = f(x)− x ≥ f(0)− x.

'Epetai ìti g(−x1) > 0. Apì to je¸rhma endi�meshc tim c up�rqei x0 ∈ (−x1, x1)o opoÐoc ikanopoieÐ thn g(x0) = 0. Dhlad , f(x0) = x0.

17. Jewr ste th suneq sun�rthsh f − g : [a, b] → R. H f − g paÐrneiel�qisth tim m se k�poio y ∈ [a, b]. Apì thn upìjesh èqoume m = (f −g)(y) =

f(y) − g(y) > 0. An jèsoume ρ = m2 , tìte ρ > 0 kai gia k�je x ∈ [a, b] èqoume

f(x)− g(x) ≥ m > ρ.

18. (a) 'Estw x ∈ (−1, 1). MporoÔme na gr�youme√

1 + x−√1− x

x− 1 =

2√1 + x +

√1− x

− 1

=1√

1 + x +√

1− x

((1−√1 + x

)+

(1−√1− x

))

=1√

1 + x +√

1− x

( −x

1 +√

1 + x+

x

1 +√

1− x

).

Parathr ste ìti√

1 + x +√

1− x > 1, 1 +√

1 + x > 1 kai 1 +√

1− x > 1.Sunep¸c,∣∣∣∣√

1 + x−√1− x

x− 1

∣∣∣∣ ≤ 1√1 + x +

√1− x

( |x|1 +

√1 + x

+|x|

1 +√

1− x

)

≤ 2|x|.

'Estw ε > 0. An 0 < |x| < δ = ε/2 tìte∣∣∣√

1+x−√1−xx − 1

∣∣∣ < ε. 'Ara,

limx→0

√1+x−√1−x

x = 1.

(b) 'Estw x > max{−a, 0}. MporoÔme na gr�youme

√x

(√x + a−√x

)− a

2=

a√

x√x + a +

√x− a

2

=a

2(√

x + a +√

x)

(√x−√x + a

)

= − a2

2(√

x + a +√

x)2.

AfoÔ (√

x + a +√

x)2 > x, èqoume

∣∣∣√x(√

x + a−√x)− a

2

∣∣∣ ≤ a2

2x.

'Estw ε > 0. An x > M = a2/(2ε) > 0 tìte∣∣√x

(√x + a−√x

)− a2

∣∣ < ε. 'Ara,

limx→+∞

√x

(√x + a−√x

)=

a

2.

19. (a) Parathr ste ìti x3−8x−2 = x2 + 2x + 4 gia k�je x 6= 2. An (xn) eÐnai mia

akoloujÐa sto R me xn 6= 2 gia k�je n ∈ N kai xn → 2, tìte x2n + 2xn + 4 →

22 + 2 · 2 + 4 = 12. Apì thn arq thc metafor�c, limx→2

x3−8x−2 = 12.

(b) An x0 /∈ Z, tìte [x0] < x0 < [x0] + 1, �ra up�rqei δ > 0 ¸ste [x0] <

x0 − δ < x0 + δ < [x0] + 1. Tìte, h f(x) = [x] eÐnai stajer kai Ðsh me [x0] semia perioq tou x0 (exhg ste giatÐ), �ra lim

x→x0[x] = [x0]. An x0 ∈ Z, tìte gia

k�je x ∈ (x0 − 1, x0) èqoume f(x) = [x] = x0 − 1 en¸ gia k�je x ∈ (x0, x0 + 1)

èqoume f(x) = [x] = x0. 'Ara, limx→x−0

[x] = x0 − 1 6= x0 = limx→x+

0

[x]. 'Epetai ìti to

limx→x0

[x] den up�rqei.

(g) Apì to (b), an x0 /∈ Z tìte limx→x0

(x− [x]) = limx→x0

x− limx→x0

[x] = x0− [x0]. An

x0 ∈ Z tìte to limx→x0

(x − [x]) den up�rqei, giatÐ tìte ja up rqe kai to limx→x0

[x]

(exhg ste giatÐ).

20. Qrhsimopoi¸ntac thn |f(x)| = |x| kai ton ε−δ orismì, deÐxte ìti limx→0

f(x) =0. Qrhsimopoi¸ntac thn puknìthta twn rht¸n kai twn arr twn deÐxte, me b�shthn arq thc metafor�c, ìti an x0 6= 0 tìte den up�rqei to lim

x→x0f(x).

21. (a) 'Estw x > 0. Parathr ste ìti bx − 1 <

[bx

] ≤ bx kai x

a > 0, �ra

b

a− x

a<

x

a

[b

x

]≤ b

a.

AfoÔ limx→0+

(ba − x

a

)= b

a , sumperaÐnoume ìti limx→0+

xa

[bx

]= b

a .

Gia to deÔtero ìrio, parathr ste ìti an 0 < x < a tìte[

xa

]=0. 'Ara,

bx

[xa

]= 0 gia k�je x ∈ (0, a). 'Epetai ìti limx→0+

bx

[xa

]= 0.

(b)'Estw x < 0. Parathr ste ìti bx − 1 <

[bx

] ≤ bx kai x

a < 0, �ra

b

a− x

a>

x

a

[b

x

]≥ b

a.

AfoÔ limx→0−

(ba − x

a

)= b

a , sumperaÐnoume ìti limx→0−

xa

[bx

]= b

a .Gia to deÔtero ìrio, parathr ste ìti an −a < x < 0 tìte −1 < x

a <

0, �ra[

xa

]= −1. Sunep¸c, b

x

[xa

]= − b

x gia k�je x ∈ (−a, 0). 'Epetai ìtilimx→0−

bx

[xa

]= +∞.

22. ArkeÐ na deÐxoume ìti h exÐswsh

g(x) = α(x− µ)(x− ν) + β(x− λ)(x− ν) + γ(x− λ)(x− µ) = 0

èqei toul�qiston mÐa rÐza se kajèna apì ta diast mata (λ, µ) kai (µ, ν). H g

eÐnai suneq c sto [λ, µ] kai sto [µ, ν]. ParathroÔme ìti

g(λ) = α(λ− µ)(λ− ν) > 0,

g(µ) = β(µ− λ)(µ− ν) < 0,

g(ν) = γ(ν − λ)(ν − µ) > 0.

Apì to je¸rhma endi�meshc tim c up�rqei ξ1 ∈ (λ, µ) ¸ste g(ξ1) = 0 kai up�rqeiξ2 ∈ (µ, ν) ¸ste g(ξ2) = 0.

23. (a) 'Estw ìti limx→x0

f(x) = α kai limx→x0

g(x) = β. Jewr ste tuqoÔsa ako-

loujÐa (xn) sto R me xn 6= x0 gia k�je n ∈ N kai xn → x0. Tìte, f(xn) ≤ g(xn)gia k�je n ∈ N, f(xn) → α kai g(xn) → β. 'Ara, α ≤ β.

(b) Jewr ste th sun�rthsh f pou orÐzetai apì thn f(x) = 0 kai th sun�rthshg pou orÐzetai apì tic g(x) = x2 an x 6= 0 kai g(0) = 5. Tìte, f(x) < g(x) giak�je x ∈ R all� lim

x→0f(x) = lim

x→0g(x) = 0 (exhg ste giatÐ).

24. Apì thn upìjesh up�rqoun δ > 0 kai M > 0 ¸ste: an x ∈ X kai |x−x0| < δ

tìte |f(x)| ≤ M . Jewr ste tuqoÔsa akoloujÐa (xn) sto X me xn 6= x0 giak�je n ∈ N kai xn → x0. Ja deÐxoume ìti f(xn)g(xn) → 0, opìte to zhtoÔmenoprokÔptei apì thn arq thc metafor�c.

'Estw ε > 0. Apì thn upìjesh èqoume limn→∞

g(xn) = 0, �ra up�rqei n1(ε) ∈ N¸ste |g(xn)| < ε

M gia k�je n ≥ n1. AfoÔ xn → x0, up�rqei n2(δ) ∈ N ¸ste:|xn − x0| < δ gia k�je n ≥ n2. 'Ara, |f(xn)| ≤ M gia k�je n ≥ n2. Jètoumen0 = max{n1, n2}. Tìte, gia k�je n ≥ n0 èqoume

|f(xn)g(xn)| = |f(xn)| · |g(xn)| < M · ε

M.

To ε > 0 tan tuqìn, �ra f(xn)g(xn) → 0.

25. 'Estw X èna fragmèno kai �peiro uposÔnolo tou R. Tìte, up�rqei ako-loujÐa (xn) sto X me ìrouc diaforetikoÔc an� dÔo: an n 6= m tìte xn 6= xm

(exhg ste giatÐ). AfoÔ to X eÐnai fragmèno, h akoloujÐa (xn) eÐnai fragmènh.Apì to je¸rhma Bolzano-Weierstrass, h (xn) èqei mia upakoloujÐa (xkn) h opoÐasugklÐnei se k�poion x0 ∈ R. Ja deÐxoume ìti o x0 eÐnai shmeÐo suss¸reushctou X deÐqnontac ìti k�je perioq tou x0 perièqei �peira to pl joc shmeÐa touX (autìc eÐnai qarakthrismìc tou shmeÐou suss¸reushc).

'Estw δ > 0. AfoÔ xkn → x0, up�rqei n0 ∈ N ¸ste: gia k�je n ≥ n0 isqÔei|xkn − x0| < δ. Tìte,

A := {xkn : n ≥ n0} ⊆ (x0 − δ, x0 + δ) ∩X.

'Omwc, to A èqei �peira to pl joc stoiqeÐa giatÐ oi xkn , n ≥ n0 eÐnai diaforetikoÐan� dÔo.

26. Jètoume ε = f(0) > 0. AfoÔ limx→+∞

f(x) = 0, up�rqei M > 0 ¸ste: gia

k�je x > M isqÔei 0 < f(x) < f(0). AfoÔ limx→−∞

f(x) = 0, up�rqei N > 0

¸ste: gia k�je x < −N isqÔei 0 < f(x) < f(0).H f eÐnai suneq c sto kleistì di�sthma [−N, M ]. 'Ara, up�rqei y ∈ [−N, M ]

me thn idiìthta: gia k�je x ∈ [−N, M ] isqÔei f(x) ≤ f(y). Eidikìtera, afoÔ−N < 0 < M èqoume f(0) ≤ f(y).

MporoÔme t¸ra eÔkola na doÔme ìti h f paÐrnei mègisth tim sto shmeÐo y.Jewr ste tuqìn x ∈ R kai diakrÐnete tic peript¸seic x < −N , x ∈ [−N, M ] kaix > M .

27. An h f eÐnai stajer kai f(x) = α gia k�je x ∈ R, tìte h f paÐrnei profan¸cmègisth kai el�qisth tim (thn α). Alli¸c, eÐte up�rqei x1 ¸ste f(x1) > α up�rqei x2 ¸ste f(x2) < α (mporeÐ fusik� na sumbaÐnoun kai ta dÔo).

Me thn upìjesh ìti up�rqei x1 ¸ste f(x1) > α, ja deÐxoume ìti h f paÐr-nei mègisth tim . H apìdeixh eÐnai entel¸c an�logh me aut n sthn prohgoÔ-menh 'Askhsh. Jètoume ε = f(x1) − α > 0. AfoÔ lim

x→+∞f(x) = α, up�rqei

M > max{0, x1} ¸ste: gia k�je x > M isqÔei f(x) < α + ε = f(x1). AfoÔlim

x→−∞f(x) = α, up�rqei N > 0 ¸ste −N < x1 kai gia k�je x < −N na isqÔei

f(x) < α + ε = f(x1). H f eÐnai suneq c sto kleistì di�sthma [−N, M ]. 'Ara,up�rqei y ∈ [−N, M ] me thn idiìthta: gia k�je x ∈ [−N,M ] isqÔei f(x) ≤ f(y).Eidikìtera, afoÔ −N < x1 < M èqoume f(x1) ≤ f(y). MporoÔme t¸ra eÔkolana doÔme ìti h f paÐrnei mègisth tim sto shmeÐo y. Jewr ste tuqìn x ∈ R kaidiakrÐnete tic peript¸seic x < −N , x ∈ [−N, M ] kai x > M .

Me thn upìjesh ìti up�rqei x2 ¸ste f(x2) < α, deÐxte ìti h f paÐrnei el�qi-sth tim .

28. O egkleismìc f(R) ⊆ R eÐnai profan c. Ja deÐxoume ìti gia k�je y ∈ Rup�rqei x ∈ R ¸ste y = f(x), opìte R ⊆ f(R).

'Estw y ∈ R. AfoÔ limx→−∞

f(x) = −∞, up�rqei x1 ∈ R ¸ste x1 < 0 kai

f(x1) < y (exhg ste giatÐ). AfoÔ limx→+∞

f(x) = +∞, up�rqei x2 ∈ R ¸ste

x2 > 0 kai f(x2) > y (exhg ste giatÐ). AfoÔ f(x1) < y < f(x2) kai h f eÐnaisuneq c sto [x1, x2], apì to je¸rhma endi�meshc tim c up�rqei x ∈ (x1, x2) ¸stef(x) = y.

29. AfoÔ h f : (α, β) → R eÐnai suneq c, gnwrÐzoume ìti to f((α, β)) eÐnai ènadi�sthma J to opoÐo perièqei to (γ, δ), ìpou γ = inf

α<x<βf(x) kai δ = sup

α<x<βf(x).

DeÐxte pr¸ta ìti limx→α+

f(x) = γ kai limx→β−

f(x) = δ. Gia par�deigma, h pr¸th

isìthta èpetai apì ta ex c:1. An α < x < β tìte f(x) ≥ γ, �ra lim

x→α+f(x) ≥ γ.

2. An α < y < β, tìte gia k�je x ∈ (α, y) èqoume f(x) < f(y), �ra limx→α+

f(x) ≤f(y). Dhlad , to lim

x→α+f(x) eÐnai k�tw fr�gma tou {f(y) : α < y < β}, opìte

limx→α+

f(x) ≤ γ.

Mènei na deÐxoume ìti to f((α, β)) den perièqei ta γ kai δ (an aut� eÐnaipeperasmèna). Autì ìmwc eÐnai sunèpeia tou ìti h f eÐnai gnhsÐwc aÔxousa: giapar�deigma, an γ = f(x) gia k�poio x ∈ (α, β) tìte paÐrnontac tuqìn α < z < x

ja eÐqame f(z) < f(x) = γ, pou eÐnai �topo afoÔ o γ eÐnai k�tw fr�gma touf((α, β)).

G. Ask seic

1. DeÐqnoume pr¸ta ìti an o x ∈ (0, 1] eÐnai rhtìc, opìte gr�fetai sth morf x = p

q ìpou p, q ∈ N me MKD(p, q) = 1, tìte h f eÐnai asuneq c sto shmeÐox. Pr�gmati, up�rqei akoloujÐa arr twn arijm¸n αn ∈ [0, 1] me αn → x. Tìte,f(αn) = 0 → 0 6= 1

q = f(x), kai to sumpèrasma èpetai apì thn arq thcmetafor�c.

'Estw t¸ra ìti o x ∈ [0, 1] eÐnai �rrhtoc kai èstw ε > 0. Jètoume M =M(ε) =

[1ε

]kai A(ε) = {y ∈ [0, 1] : f(y) ≥ ε}. An o y an kei sto A(ε) tìte eÐnai

rhtìc o opoÐoc gr�fetai sth morf x = pq ìpou p, q ∈ N, p ≤ q kai f(y) = 1

q ≥ ε.To pl joc aut¸n twn arijm¸n eÐnai to polÔ Ðso me to pl joc twn zeugari¸n(p, q) fusik¸n arijm¸n ìpou q ≤ M kai p ≤ q. Epomènwc, den xepern�ei tonM(M + 1)/2. Dhlad , to A(ε) eÐnai peperasmèno sÔnolo. MporoÔme loipìn nagr�youme A(ε) = {y1, . . . , ym} ìpou m = m(ε) ∈ N.

AfoÔ o x eÐnai �rrhtoc, o x den an kei sto A(ε). 'Ara, o arijmìc δ = min{|x−y1|, . . . , |x − ym|} eÐnai gn sia jetikìc. 'Estw z ∈ [0, 1] me |z − x| < δ. Tìte,z /∈ A(ε) �ra f(z) < ε. AfoÔ f(x) = 0, èpetai ìti 0 ≤ f(z) = f(z) − f(x) < ε.To ε > 0 tan tuqìn, �ra h f eÐnai suneq c sto shmeÐo x.

Tèloc, deÐxte ìti h f eÐnai suneq c sto shmeÐo 0.

2. (⇐) Ac upojèsoume ìti h f den eÐnai suneq c sto x0. Up�rqei ε > 0 ¸ste:gia k�je n ∈ N mporoÔme na broÔme xn ∈ (a, b) gia to opoÐo |xn − x0| < 1

n

kai |f(xn)− f(x0)| ≥ ε (exhg ste giatÐ). GnwrÐzoume ìti k�je akoloujÐa prag-matik¸n arijm¸n èqei monìtonh upakoloujÐa. 'Ara, up�rqei monìtonh upako-loujÐa (xkn) thc (xn) h opoÐa ikanopoieÐ ta ex c: |xkn − x0| < 1

kn≤ 1

n kai|f(xkn)− f(x0)| ≥ ε gia k�je n ∈ N. Apì thn |xkn − x0| < 1

n èqoume xkn → x0.Apì thn |f(xkn)− f(x0)| ≥ ε èqoume f(xkn) 6→ f(x0). Autì eÐnai �topo apì thnupìjesh.

H kateÔjunsh (⇒) prokÔptei �mesa apì thn arq thc metafor�c.

3. Apì thn upìjesh èpetai ìti f(0) = f(

mn

)gia k�je m ∈ Z kai k�je n ∈ N

(exhg ste giatÐ). 'Estw x ∈ R. An mn(x) = [nx], tìte mn(x) ∈ Z kai mn(x) ≤nx < mn(x) + 1. 'Ara,

mn(x)n

≤ x <mn(x)

n+

1n

.

Parathr ste ìti f(

mn(x)n

)= f(0) gia k�je n ∈ N kai ìti x = lim

n→∞mn(x)

n . Apìth sunèqeia thc f sto shmeÐo x sumperaÐnoume ìti

f(x) = limn→∞

f

(mn(x)

n

)= f(0).

To x tan tuqìn, �ra h f eÐnai stajer .

4. OrÐzoume g :[0, 1− 1

n

] → R me g(x) = f(x) − f(x + 1

n

). H g eÐnai kal�

orismènh kai suneq c sto[0, 1− 1

n

]. Parathr ste ìti

g(0) + g

(1n

)+ · · ·+ g

(n− 1

n

)= f(0)− f(1) = 0.

'Ara, up�rqoun κ, λ ∈ {0, 1, . . . , n − 1} ¸ste g(

κn

) ≤ 0 kai g(

λn

) ≥ 0. Apì toje¸rhma endi�meshc tim c up�rqei x pou an kei sto kleistì di�sthma pou èqei�kra ta κ

n kai λn kai ikanopoieÐ thn f(x)− f

(x + 1

n

)= g(x) = 0.

5. Apì to je¸rhma stajeroÔ shmeÐou (deÐte kai thn 'Askhsh 15) gnwrÐzoume ìtiup�rqei x1 ∈ [0, 1] ¸ste g(x1) = x1. OrÐzoume anadromik� mia akoloujÐa (xn)sto [0, 1] jètontac xn+1 = f(xn) gia k�je n ∈ N. Parathr ste ìti g(xn) = xn

gia k�je n ∈ N. Pr�gmati, autì isqÔei gia n = 1 kai an g(xm) = xm tìte,qrhsimopoi¸ntac thn f ◦ g = g ◦ f èqoume

g(xm+1) = g(f(xm)) = (g ◦f)(xm) = (f ◦g)(xm) = f(g(xm)) = f(xm) = xm+1.

An gia k�poio n ∈ N èqoume f(xn) = xn tìte to xn eÐnai koinì stajerì shmeÐotwn f kai g.

Upojètoume loipìn ìti xn+1 = f(xn) 6= xn gia k�je n ∈ N. Eidikìtera,x2 = f(x1) 6= x1. QwrÐc periorismì thc genikìthtac upojètoume ìti x2 > x1 (anx2 < x1 douleÔoume me ton Ðdio trìpo). Tìte, x3 = f(x2) > f(x1) = x2 (giatÐh f eÐnai aÔxousa kai èqoume upojèsei ìti xn+1 6= xn gia k�je n). Epagwgik�deÐqnoume ìti h (xn) eÐnai gnhsÐwc aÔxousa. AfoÔ xn ≤ 1 gia k�je n ∈ N, èpetaiìti xn → x0 gia k�poio x0 ∈ [0, 1]. H sunèqeia thc f sto x0 deÐqnei ìti

f(x0) = limn→∞

f(xn) = limn→∞

xn+1 = x0.

Apì thn �llh pleur�, h sunèqeia thc g sto x0 deÐqnei ìti

g(x0) = limn→∞

g(xn) = limn→∞

xn = x0.

'Ara, to x0 eÐnai koinì stajerì shmeÐo twn f kai g.

6. Upojètoume ìti h f den eÐnai stajer . Tìte, up�rqei x ∈ R ¸ste f(x) 6=b (giatÐ?). Jewr ste thn akoloujÐa xn = x + nT . Tìte, xn → +∞ kaif(xn) = f(x) gia k�je n ∈ N (exhg ste giatÐ). 'Ara, lim

n→∞f(xn) = f(x).

AfoÔ limx→+∞

f(x) = b, apì thn arq thc metafor�c paÐrnoume limn→∞

f(xn) = b.

'Ara f(x) = b, to opoÐo eÐnai �topo.

7. 'Estw f : [0, 1] → R suneq c sun�rthsh. kai èstw xn ∈ [0, 1] ¸ste f(xn) →0. Apì to je¸rhma Bolzano–Weierstrass up�rqei upakoloujÐa (xkn) thc (xn) mexkn → x0 ∈ [0, 1]. AfoÔ f(xn) → 0, èqoume f(xkn) → 0. AfoÔ xkn → x0 kai hf eÐnai suneq c sto x0, èqoume f(xkn) → f(x0). Apì th monadikìthta tou orÐouakoloujÐac, f(x0) = 0.

8. Apì thn an → a kai apì thn arq thc metafor�c, èqoume f(an) → f(a). Apìthn upìjesh, f(an) = an+1 → a. Apì th monadikìthta tou orÐou akoloujÐac,f(a) = a.

9. 'Estw ìti h f den eÐnai fragmènh. Tìte, mporoÔme na broÔme xn ∈ [a, b] ¸ste|f(x1)| > 1 kai |f(xn)| > max{n, f(xn−1)} gia k�je n ≥ 2. Eidikìtera, oi ìroithc (xn) eÐnai diaforetikoÐ an� dÔo.

Apì to je¸rhma Bolzano–Weierstrass up�rqei upakoloujÐa (xkn) thc (xn) mexkn → x0 ∈ [0, 1]. Apì thn kataskeu , oi ìroi thc (xkn) eÐnai telik� diaforetikoÐapì ton x0 (exhg ste giatÐ).

Apì thn upìjesh, up�rqei to limx→x0

f(x) = m. H arq thc metafor�c deÐqnei

ìti f(xkn) → m, �ra h (f(xkn)) eÐnai fragmènh. 'Omwc, |f(xkn | > kn ≥ n. 'Ara,|f(xkn

)| → +∞. 'Etsi, katal goume se �topo.

10. DeÐqnoume pr¸ta ìti an [c, d] ⊆ [a, b] kai c < x < d tìte to f(x) an kei stokleistì di�sthma pou èqei �kra ta f(c) kai f(d). Gia to skopì autì, mporoÔmena upojèsoume ìti f(c) ≤ f(d) (sthn antÐjeth perÐptwsh, douleÔoume an�loga).H f paÐrnei mègisth tim sto [c, d] se k�poio shmeÐo x0. An f(x0) > f(d) tìtec < x0 < d, �ra up�rqei δ > 0 ¸ste (x0 − δ, x0 + δ) ⊂ [c, d]. Tìte, gia k�jex ∈ (x0 − δ, x0 + δ) èqoume f(x) ≤ f(x0), dhlad h f èqei topikì mègisto stox0. Autì eÐnai �topo, �ra h mègisth tim thc f sto [c, d] eÐnai h f(d). 'Omoia,h f paÐrnei el�qisth tim sto [c, d] se k�poio shmeÐo x1. An f(x1) < f(c) tìtec < x1 < d, �ra up�rqei δ > 0 ¸ste (x1 − δ, x1 + δ) ⊂ [c, d]. Tìte, gia k�jex ∈ (x1 − δ, x1 + δ) èqoume f(x) ≥ f(x1), dhlad h f èqei topikì el�qisto stox1. Autì eÐnai �topo, �ra h el�qisth tim thc f sto [c, d] eÐnai h f(c). Dhlad ,gia k�je x ∈ [c, d] èqoume

(∗) f(c) ≤ f(x) ≤ f(d).

T¸ra mporoÔme na deÐxoume ìti h f eÐnai monìtonh. MporoÔme na upojèsoume ìtif(a) ≤ f(b) (sthn antÐjeth perÐptwsh, douleÔoume an�loga). Ja deÐxoume ìtih f eÐnai aÔxousa. 'Estw a < x < y < b. Efarmìzontac thn (∗) gia thn tri�daa, x, b èqoume f(a) ≤ f(x) ≤ f(b). Efarmìzontac thn (∗) gia thn tri�da x, y, b

èqoume f(x) ≤ f(y) ≤ f(b). 'Ara, gia k�je x < y sto (a, b) èqoume f(x) ≤ f(y).'Epetai ìti h f eÐnai aÔxousa sto [a, b]. MporoÔme m�lista na doÔme ìti h f eÐnaignhsÐwc aÔxousa. An up rqan x < y sto [a, b] me f(x) = f(y) tìte h f ja tanstajer sto [x, y]. 'Omwc tìte, h f ja eÐqe topikì mègisto kai topikì el�qistose k�je z ∈ (x, y).

11. Upojètoume ìti h f èqei topikì mègisto se dÔo shmeÐa x1 < x2 tou [a, b]. H f

eÐnai suneq c sto [x1, x2], �ra up�rqei x3 ∈ [x1, x2] me thn idiìthta: f(x3) ≤ f(x)gia k�je x ∈ [x1, x2]. DiakrÐnoume treic peript¸seic:(a) x1 < x3 < x2: Tìte, an p�roume δ = min{x3 − x1, x2 − x3} > 0 èqoume(x3− δ, x3 + δ) ⊂ [x1, x2] ⊆ [a, b] kai f(x3) ≤ f(x) gia k�je x ∈ (x3− δ, x3 + δ).'Ara, h f èqei topikì el�qisto sto x3.(b) x1 = x3: Tìte, up�rqei 0 < δ1 < x2 − x1 ¸ste f(x) ≤ f(x3) = f(x1)gia k�je x1 < x < x1 + δ1 (giatÐ h f èqei topikì mègisto sto x1) kai up�rqei0 < δ2 < x2−x1 ¸ste f(x) ≥ f(x3) = f(x1) gia k�je x1 < x < x1+δ2 (giatÐ h f

èqei topikì el�qisto sto x3 = x1). 'Epetai ìti: an jèsoume δ = min{δ1, δ2} > 0tìte h f eÐnai stajer sto (x1, x1 + δ). 'Ara, h f èqei topikì el�qisto se k�jeshmeÐo tou (x1, x1 + δ).(g) x2 = x3: 'Omoia me to (b).

6.6 Par�gwgoc kai melèth sunart sewn

6.6 Paragwgoc kai meleth sunarthsewn · 193

A1. Exet�ste an oi parak�tw prot�seic eÐnai alhjeÐc yeudeÐc (aitiolog stepl rwc thn ap�nthsh sac).

1. Swstì. 'Estw x ∈ (a, b). Apì thn upìjesh, h f eÐnai paragwgÐsimh sto x,�ra eÐnai suneq c sto x.

2. Swstì. AfoÔ f(0) = f ′(0) = 0, èqoume

0 = f ′(0) = limx→0

f(x)− f(0)x

= limx→0

f(x)x

.

Apì thn arq thc metafor�c gia to ìrio, an xn 6= 0 kai xn → 0, tìte limn→∞

f(xn)xn

=

0. Jewr¸ntac thn akoloujÐa xn = 1n → 0, paÐrnoume lim

n→∞nf(1/n) = lim

n→∞f(1/n)

1/n =0.

3. L�joc. Jewr ste th sun�rthsh f : [0, 1] → R me f(x) = 1 − x. H f

eÐnai paragwgÐsimh sto [0, 1] kai paÐrnei th mègisth tim thc sto x0 = 0, ìmwcf ′(x) = −1 gia k�je x ∈ [0, 1], �ra f ′(0) = −1 6= 0.

4. Swstì. 'Estw x > 0. Efarmìzoume to je¸rhma mèshc tim c sto [0, x]:up�rqei ξx ∈ (0, x) ¸ste

f(x) = f(x)− f(0) = xf ′(ξx).

AfoÔ x > 0 kai f ′(ξ) ≥ 0, sumperaÐnoume ìti f(x) ≥ 0.Gia x = 0, èqoume f(x) = f(0) = 0.

5. Swstì. Efarmìzontac to je¸rhma Rolle gia thn f sta [0, 1] kai [1, 2], brÐ-skoume y1 ∈ (0, 1) me f ′(y1) = 0 kai y2 ∈ (1, 2) me f ′(y2) = 0. Efarmìzontac p�lito je¸rhma Rolle gia thn f ′ sto [y1, y2], brÐskoume x0 ∈ (y1, y2) me f ′′(x0) = 0.Tèloc, 0 < y1 < x0 < y2 < 2, dhlad x0 ∈ (0, 2).

6. Swstì. ArkeÐ na deÐxoume ìti limx→x0

f(x)−f(x0)x−x0

= `. 'Estw ε > 0. AfoÔ

limx→x0

f ′(x) = ` ∈ R, up�rqei δ > 0 ¸ste: an 0 < |y−x0| < δ, tìte |f ′(y)−`| < ε.

'Estw x ∈ (a, b) me x0 < x < x0 + δ. Apì tic upojèseic mac èpetai ìti f

eÐnai suneq c sto [x0, x] kai paragwgÐsimh sto (x0, x), opìte, efarmìzontac toje¸rhma mèshc tim c sto [x0, x], brÐskoume yx ∈ (x0, x) ¸ste f(x)−f(x0)

x−x0=

f ′(yx). 'Omwc, 0 < |yx − x0| < |x− x0| < δ, �ra |f ′(yx)− `| < ε. Sunep¸c,

(∗)∣∣∣∣f(x)− f(x0)

x− x0− `

∣∣∣∣ = |f ′(yx)− `| < ε.

AfoÔ to ε > 0 tan tuqìn kai h (∗) isqÔei gia k�je x ∈ (x0, x0+δ), sumperaÐnoumeìti lim

x→x+0

f(x)−f(x0)x−x0

= `. Me ton Ðdio trìpo deÐqnoume ìti to ìrio apì arister�

isoÔtai me `, �ra h f eÐnai paragwgÐsimh sto x0, kai f ′(x0) = `.

7. L�joc. JewroÔme th sun�rthsh f : R → R me f(x) = x2 an x ∈ Q kaif(x) = −x2 an x /∈ Q. H f eÐnai asuneq c se k�je x 6= 0, �ra den up�rqei δ > 0

¸ste h f na eÐnai suneq c sto (−δ, δ). 'Omwc, h f eÐnai paragwgÐsimh sto 0:èqoume ∣∣∣∣

f(x)− f(0)x

∣∣∣∣ =|f(x)||x| =

|x2||x| = |x| → 0 ìtan x → 0,

�ra f ′(0) = 0.

8*. L�joc. QrhsimopoioÔme mia parallag tou prohgoÔmenou paradeÐgmatoc.JewroÔme th sun�rthsh f : R→ R me f(x) = x+x2 an x ∈ Q kai f(x) = x−x2

an x /∈ Q. H f eÐnai paragwgÐsimh sto 0 kai f ′(0) = 1 > 0 (exhg ste giatÐ). JadeÐxoume ìmwc ìti, gia k�je δ > 0, h f den eÐnai aÔxousa sto (0, δ). Jewr stetuqìnta rhtì x ∈ (0, δ). Apì thn puknìthta twn arr twn sto R, up�rqei �rrhtocy ¸ste x < y < min{δ, x + x2}. Tìte, y ∈ (0, δ) kai x < y, ìmwc

f(y) = y − y2 < y < x + x2 = f(x).

'Ara, h f den eÐnai aÔxousa sto (0, δ).

A2. D¸ste par�deigma sun�rthshc f : R→ R me tic ex c idiìthtec:(a) f(−1) = 0, f(2) = 1 kai f ′(1) > 0: Jewr ste thn f : R→ R me f(x) = x+1

3

(th grammik sun�rthsh me f(−1) = 0 kai f(2) = 1). 'Eqoume f ′(x) = 13 > 0 gia

k�je x ∈ R, �ra, f ′(1) > 0.(b) f(−1) = 0, f(2) = 1 kai f ′(1) < 0: Jewr ste sun�rthsh thc morf cf(x) = ax2 + bx + c. Zht�me: f(−1) = a− b + c = 0 kai f(2) = 4a + 2b + c = 1,�ra c = 1

3 − 2a kai b = 13 − a. EpÐshc, f ′(x) = 2ax + b, �ra

f ′(1) = 2a + b = a +13

< 0 an a < −13.

T¸ra, mporoÔme na epilèxoume: a = − 23 , b = 1, c = 5

3 . Elègxte ìti h sun�rthshf(x) = − 2

3x2 + x + 53 ikanopoieÐ to zhtoÔmeno.

(g) f(0) = 0, f(3) = 1, f ′(1) = 0 kai h f eÐnai gnhsÐwc aÔxousa sto [0, 3]: Totupikì par�deigma gnhsÐwc aÔxousac paragwgÐsimhc sun�rthshc pou h par�gw-gìc thc mhdenÐzetai se èna shmeÐo eÐnai h g(x) = x3. Jewr ste sun�rthsh thcmorf c f(x) = a(x − 1)3 + b. Zht�me: f(0) = −a + b = 0, �ra b = a. EpÐshc,f(3) = 8a + a = 1, �ra a = 1

9 . Elègxte ìti h sun�rthsh f(x) = (x−1)3+19

ikanopoieÐ to zhtoÔmeno.(d) f(m) = 0 kai f ′(m) = (−1)m gia k�je m ∈ Z, |f(x)| ≤ 1

2 gia k�je x ∈ R:Jewr ste th sun�rthsh f(x) = a sin(πx). Tìte, f(m) = 0 gia k�je m ∈ Z kaif ′(m) = πa cos(πm) = (−1)mπa gia k�je m ∈ Z. Prèpei loipìn na epilèxoumea = 1

π ¸ste na ikanopoioÔntai oi dÔo pr¸tec sunj kec. Tìte,

|f(x)| =∣∣∣∣1π

sin(πx)∣∣∣∣ ≤

1π≤ 1

2

gia k�je x ∈ R, afoÔ π > 2. Dhlad , ikanopoieÐtai kai h trÐth sunj kh.

1. (a) f(x) = x3 − x2 − 8x + 1 sto [−2, 2]. H f eÐnai paragwgÐsimh sto (−2, 2)kai f ′(x) = 3x2 − 2x − 8. Oi rÐzec thc parag¸gou eÐnai: x1 = 2 kai x2 = − 4

3 .'Ara, to monadikì krÐsimo shmeÐo thc f sto (−2, 2) eÐnai to x2. UpologÐzoume tictimèc

f(−2) = 5, f(2) = −11, f(−4/3) = 203/27.

'Epetai ìti max(f) = 203/27 kai min(f) = −11.

(b) f(x) = x5 + x + 1 sto [−1, 1]. H f eÐnai paragwgÐsimh sto (−1, 1) kaif ′(x) = 5x4 + 1 > 0, dhlad h f den èqei krÐsima shmeÐa. UpologÐzoume tic timècf(−1) = −1 kai f(1) = 3. 'Epetai ìti max(f) = 3 kai min(f) = −1.

(g) f(x) = x3−3x sto [−1, 2]. H f eÐnai paragwgÐsimh sto (−1, 2), me par�gwgof ′(x) = 3x2− 3. Ta shmeÐa sta opoÐa mhdenÐzetai h par�gwgoc eÐnai ta x1 = −1kai x2 = 1. 'Ara, to monadikì krÐsimo shmeÐo thc f sto (−1, 2) eÐnai to x2.UpologÐzoume tic timèc

f(−1) = 2, f(1) = −2, f(2) = 2.

'Epetai ìti max(f) = 2 kai min(f) = −2.

2. (a) H exÐswsh 4ax3+3bx2+2cx = a+b+c èqei toul�qiston mÐa rÐza sto (0, 1).Jewr ste th sun�rthsh f : [0, 1] → R me f(x) = ax4 + bx3 + cx2−ax− bx− cx.Parathr ste ìti f(0) = f(1) = 0. Efarmìzontac to je¸rhma Rolle brÐskoumemia rÐza thc exÐswshc sto (0, 1).

(b) H exÐswsh 6x4 − 7x + 1 = 0 èqei to polÔ dÔo pragmatikèc rÐzec. JewroÔmeth sun�rthsh f(x) = 6x4 − 7x + 1. Ac upojèsoume ìti up�rqoun x1 < x2 < x3

¸ste f(x1) = f(x2) = f(x3) = 0 (dhlad , ìti h exÐswsh èqei perissìterec apìdÔo pragmatikèc rÐzec). Efarmìzontac to je¸rhma Rolle gia thn f sta [x1, x2]kai [x2, x3], brÐskoume x1 < y1 < x2 < y2 < x3 ¸ste f ′(y1) = f ′(y2) = 0.Dhlad , h exÐswsh f ′(x) = 0 èqei toul�qiston dÔo (diaforetikèc) pragmatikècrÐzec. 'Omwc, f ′(x) = 24x3 − 7 = 0 an kai mìno an x = 3

√7/24, dhlad h

f ′(x) = 0 èqei akrib¸c mÐa pragmatik rÐza. Katal xame se �topo, �ra h arqik exÐswsh èqei to polÔ dÔo pragmatikèc rÐzec.

(g) H exÐswsh x3+9x2+33x−8 = 0 èqei akrib¸c mÐa pragmatik rÐza. JewroÔmeth sun�rthsh f(x) = x3+9x2+33x−8. AfoÔ f(0) = −8 < 0 kai f(1) = 35 > 0,apì to je¸rhma endi�meshc tim c sumperaÐnoume ìti h exÐswsh f(x) = 0 èqeitoul�qiston mÐa rÐza sto (0, 1).

An upojèsoume ìti h f(x) = 0 èqei dÔo diaforetikèc pragmatikèc rÐzec, tìteh f ′(x) = 0 èqei toul�qiston mÐa pragmatik rÐza (je¸rhma Rolle). 'Omwc,f ′(x) = 3x2 + 18x + 33 = 3(x2 + 6x + 11) > 0 gia k�je x ∈ R (elègxte ìti hdiakrÐnousa tou triwnÔmou eÐnai arnhtik ). Autì eÐnai �topo, �ra h f(x) = 0 èqeito polÔ mÐa pragmatik rÐza.

Apì ta parap�nw, h f(x) = 0 èqei akrib¸c mÐa pragmatik rÐza.

3. JewroÔme thn f(x) = xn + ax + b kai upojètoume pr¸ta ìti o n ≥ 4 eÐnai�rtioc (gia n = 2 den èqoume tÐpota na deÐxoume). 'Estw ìti h f(x) = 0 èqei treÐc

diaforetikèc pragmatikèc rÐzec. Apì to je¸rhma Rolle, h f ′(x) = nxn−1 +a = 0èqei toul�qiston dÔo diaforetikèc pragmatikèc rÐzec. Autì eÐnai �topo: o n− 1eÐnai perittìc, �ra nxn−1 + a = 0 an kai mìno an x = n−1

√−a/n (monadik

pragmatik rÐza).'Estw t¸ra ìti o n ≥ 3 eÐnai perittìc. Tìte, o n − 1 eÐnai �rtioc kai h

nxn−1 + a = 0 èqei to polÔ dÔo rÐzec: tic ± n−1√−a/n an a < 0, thn x = 0 an

a = 0, kamÐa an a > 0. 'Ara, h f(x) = 0 èqei to polÔ treÐc pragmatikèc rÐzec(exhg ste giatÐ, qrhsimopoi¸ntac to je¸rhma Rolle).

4. An upojèsoume ìti h f ′(x) = 0 èqei n diaforetikèc pragmatikèc rÐzec, tìteefarmìzontac to je¸rhma Rolle blèpoume ìti h f ′′(x) = 0 èqei n−1 diaforetikècpragmatikèc rÐzec, kai, suneqÐzontac me ton Ðdio trìpo, ìti h f (n)(x) = 0 èqei(toul�qiston) mÐa pragmatik rÐza. 'Omwc, f (n)(x) = n! 6= 0 gia k�je x ∈ R, kaikatal goume se �topo.

'Ara, h f ′(x) = 0 èqei to polÔ (n− 1) pragmatikèc rÐzec. ParathroÔme t¸raìti: gia k�je i = 1, . . . , n − 1 èqoume f(ai) = f(ai+1) = 0, opìte to je¸rhmaRolle deÐqnei ìti up�rqei yi ∈ (ai, ai+1) ¸ste f ′(yi) = 0. Ta y1, . . . , yn−1 eÐnaidiaforetik� an� dÔo giatÐ ta (ai, ai+1) eÐnai xèna an� dÔo (diadoqik�) diast mata.'Ara, h f ′(x) = 0 èqei toul�qiston n− 1 pragmatikèc rÐzec.

Sundu�zontac ta parap�nw, sumperaÐnoume ìti h f ′(x) = 0 èqei akrib¸c n−1pragmatikèc rÐzec.

6. (a) Jèloume na megistopoi soume thn posìthta ab me thn upìjesh a2 + b2 =d2, ìpou d > 0 (exhg ste giatÐ). Parathr ste ìti

ab ≤ a2 + b2

2=

d2

2

me isìthta an kai mìno an a = b (= d/√

2).'Alloc trìpoc. Jewr ste th sun�rthsh f : [0, d] → R me f(a) = a

√d2 − a2 ,

isodÔnama, thn g = f2 : [0, d] → R me g(a) = a2(d2−a2). Jèloume na deÐxoume ìtimax(g) = f(d/

√2) (exhg ste giatÐ). Paragwgizontac, èqoume g′(a) = 2ad2 −

4a3 = 2a(d2 − 2a2). 'Epetai ìti h g eÐnai gnhsÐwc aÔxousa sto [0, d/√

2] kaignhsÐwc fjÐnousa sto [d/

√2, d], �ra paÐrnei th mègisth tim thc an kai mìno an

a = d/√

2.

(b) Jèloume na megistopoi soume thn posìthta ab me thn upìjesh a + b = d,ìpou d > 0 (exhg ste giatÐ). Parathr ste ìti

ab ≤ (a + b)2

4=

d2

4

me isìthta an kai mìno an a = b (= d/2).'Alloc trìpoc. Jewr ste th sun�rthsh g : [0, d] → R me g(a) = a(d − a).Jèloume na deÐxoume ìti max(g) = f(d/2) (exhg ste giatÐ). Paragwgizontac,èqoume g′(a) = d − 2a. 'Epetai ìti h g eÐnai gnhsÐwc aÔxousa sto [0, d/2] kaignhsÐwc fjÐnousa sto [d/2, d], �ra paÐrnei th mègisth tim thc an kai mìno ana = d/2.

7. 'Estw (x, y) èna shmeÐo thc uperbol c x2 − y2 = 1. To tetr�gwno thcapìstashc tou (x, y) apì to (0, 1) isoÔtai me x2 + (y − 1)2 = 1 + y2 + (y −1)2 = 2y2 − 2y + 2. Parathr ste ìti gia k�je y ∈ R up�rqoun dÔo timèctou x (oi ±

√1 + y2) ¸ste to (x, y) na an kei sthn uperbol . ArkeÐ loipìn

(exhg ste giatÐ) na broÔme thn el�qisth tim thc sun�rthshc g : R → R meg(y) = 2y2 − 2y + 2. ParagwgÐzontac, blèpoume ìti to min(g) pi�netai ìtany = 1/2, opìte paÐrnoume dÔo shmeÐa, ta

(±√

52 , 1

2

).

8. MporoÔme na upojèsoume ìti A = (1, 0) kai B = (−1, 0). ArkeÐ na jewr -soume shmeÐa Γ thc morf c (cosx, sin x), ìpou 0 ≤ x ≤ π. Aut� eÐnai ta shmeÐatou �nw hmikuklÐou, gia to k�tw hmikÔklio ergazìmaste an�loga. Parathr steìti to m koc tou AΓ eÐnai 2 sin x

2 kai to m koc tou BΓ eÐnai 2 sin π−x2 = 2 cos x

2 .ArkeÐ loipìn na megistopoi soume thn g : [0, π] → R me g(x) = sin x

2 + cos x2

(exhg ste giatÐ). AfoÔ g′(x) = 12 cos x

2 − 12 sin x

2 , sumperaÐnoume ìti h g paÐrneimègisth tim ìtan x

2 = π4 , dhlad x = π/2. 'Ara, ta zhtoÔmena shmeÐa eÐnai ta

Γ1 = (0, 1) kai Γ2 = (−1, 0).

9. Parathr ste ìti

f(x) =n∑

k=1

(x2 − 2akx + a2k) = nx2 − 2(a1 + · · ·+ an)x + (a2

1 + · · ·+ a2n).

H par�gwgoc thc f eÐnai h

f ′(x) = 2nx− 2(a1 + · · ·+ an).

'Epetai ìti h f paÐrnei thn el�qisth tim thc sto x0 = 1n (a1 + · · · + an). H

el�qisth tim eÐnai Ðsh me min(f) = (a21 + · · ·+ a2

n)− 1n (a1 + · · ·+ an)2.

10. Melet ste thn f qwrist� sta diast mata (−∞, 0], [0, a] kai [a,+∞) (¸stena {di¸xete} tic apìlutec timèc). ParagwgÐzontac, elègxte ìti h f eÐnai aÔxousasto (−∞, 0], fjÐnousa sto [a,+∞), en¸ sto [0, a] èqoume ìti h f eÐnai fjÐnousasto [0, a/2] kai aÔxousa sto [a/2, a].

Sunep¸c, h mègisth tim thc f eÐnai mÐa apì tic f(0) kai f(a). Parathr steìti f(0) = 1 + 1

1+a = 2+a1+a = f(a). Sunep¸c, max(f) = 2+a

1+a .

11. 'Eqoume f ′(x0) = limx→x0

f(x)−f(x0)x−x0

> 0. Efarmìzontac ton orismì tou orÐou

me ε = f ′(x0)2 > 0, blèpoume ìti up�rqei δ > 0 ¸ste: an 0 < |x − x0| < δ, tìte

x ∈ (a, b) kaif(x)− f(x0)

x− x0> 0.

Katìpin, diakrÐnoume tic peript¸seic x ∈ (x0, x0 + δ) kai x ∈ (x0 − δ, x0). Giapar�deigma, an x0 < x < x0 + δ, tìte

f(x)− f(x0) =f(x)− f(x0)

x− x0(x− x0) > 0, �ra f(x) > f(x0).

12. (a) AfoÔ h f èqei topikì el�qisto sto x0, èqoume f ′(x0) = 0 (to x0 eÐnaikrÐsimo shmeÐo thc f). An eÐqame f ′′(x0) < 0, tìte h f ja eÐqe topikì mègisto stox0, kai m�lista gn sio, me thn ex c ènnoia: ja up rqe δ > 0 ¸ste f(x) < f(x0)gia k�je x ∈ (x0 − δ, x0) ∪ (x0, x0 + δ) (èqei apodeiqjeÐ sth jewrÐa). Autì eÐnai�topo, afoÔ upojèsame ìti h f èqei topikì el�qisto sto x0. 'Ara, f ′′(x0) ≥ 0.

(b) Me ton Ðdio trìpo (apagwg se �topo).

13. Jèloume na deÐxoume ìti up�rqei x ∈ (a, b) ¸ste f ′(x) = g′(x) (autì shmaÐneiìti oi efaptìmenec twn grafik¸n parast�sewn twn f kai g sta (x, f(x)) kai(x, g(x)) eÐnai par�llhlec tautÐzontai). JewroÔme th sun�rthsh h = f − g :[a, b] → R. AfoÔ f(a) = g(a) kai f(b) = g(b), èqoume h(a) = h(b) = 0.Efarmìzontac to je¸rhma Rolle, brÐskoume x ∈ (a, b) ¸ste h′(x) = 0, dhlad ,f ′(x)− g′(x) = 0.

14. Upojètoume ìti up�rqoun x1 < x2 sto (a, b) ¸ste f(x1) = f(x2) = 0,kai ìti h g den mhdenÐzetai sto (x1, x2) (apagwg se �topo). Efarmìzontac thnupìjesh f(x)g′(x)− f ′(x)g(x) 6= 0 sta x1 kai x2, blèpoume ìti f ′(x1)g(x1) 6= 0kai f ′(x2)g(x2) 6= 0, �ra h g den mhdenÐzetai sta x1, x2. Me �lla lìgia, h g denmhdenÐzetai sto [x1, x2].

Tìte, mporoÔme na orÐsoume thn h := fg : [x1, x2] → R. H h eÐnai suneq c

sto [x1, x2], paragwgÐsimh sto (x1, x2), kai h(x1) = h(x2) = 0 (exhg ste giatÐ).Apì to je¸rhma Rolle, up�rqei x ∈ (x1, x2) ¸ste

h′(x) =f ′(x)g(x)− f(x)g′(x)

[g(x)]2= 0.

Autì eÐnai �topo: afoÔ x ∈ (a, b), èqoume f(x)g′(x)−f ′(x)g(x) 6= 0, �ra h′(x) 6=0.

15. Jètoume γ = a+b2 . Efarmìzontac to je¸rhma mèshc tim c sta [a, γ] kai

[γ, b] brÐskoume x1 ∈ (a, γ) kai x2 ∈ (γ, b) pou ikanopoioÔn tic

f ′(x1) =f(γ)− f(a)

γ − akai f ′(x2) =

f(b)− f(γ)b− γ

.

Qrhsimopoi¸ntac thn γ − a = b−a2 = b − γ kai thn f(a) = f(b), elègxte ìti

f ′(x1) + f ′(x2) = 0.

16. 'Estw ε > 0. AfoÔ limy→+∞

f ′(y) = 0, up�rqei M > 0 ¸ste: gia k�je y > M

isqÔei |f ′(y)| < ε. 'Estw x > M . Efarmìzoume to je¸rhma mèshc tim c stodi�sthma [x, x + 1]: up�rqei yx ∈ (x, x + 1) ¸ste

f(x + 1)− f(x) = f ′(yx)((x + 1)− x) = f ′(yx).

'Omwc yx > x > M , �ra |f ′(yx)| < ε. Dhlad ,

|f(x + 1)− f(x)| < ε.

'Epetai ìti limx→+∞

(f(x + 1)− f(x)) = 0.

17. 'Estw x > 1. Efarmìzoume to je¸rhma mèshc tim c sto di�sthma [x, x+√

x]:up�rqei yx ∈ (x, x +

√x) ¸ste

f(x +√

x)− f(x) = f ′(yx)√

x.

'Omwc yx > x > 1, �ra |f ′(yx)| ≤ 1yx

< 1x . Dhlad ,

|f(x +√

x)− f(x)| < 1x· √x =

1√x

.

'Epetai ìti limx→+∞

(f(x +√

x)− f(x)) = 0.

18. (a) H par�gwgoc thc h(x) = f(x)x sto x ∈ (0, a) isoÔtai me

h′(x) =f ′(x)x− f(x)

x2.

Efarmìzontac to je¸rhma mèshc tim c sto di�sthma [0, x] gia thn f , brÐskoumeξ ∈ (0, x) ¸ste

f(x) = f(x)− f(0) = f ′(ξ)x.

'Omwc h f ′ eÐnai aÔxousa kai ξ < x, �ra f ′(ξ) ≤ f ′(x). Sunep¸c, f(x) ≤ f ′(x)x.'Epetai ìti h′ ≥ 0 sto (0, a), �ra h h eÐnai aÔxousa.(b) H sun�rthsh h(x) = f(x)

g(x) eÐnai kal� orismènh sto (0, a). Pr�gmati, parath-

r ste ìti h f ′

g′ orÐzetai kal� sto (0, a) kai ìti g′ > 0 (apì thn upìjesh). Autìèqei san sunèpeia kai thn g(x) > 0 sto (0, a) (deÐte thn er¸thsh katanìhshc 4).'Eqoume

h′(x) =f ′(x)g(x)− g′(x)f(x)

[g(x)]2.

Efarmìzontac to je¸rhma mèshc tim c sto di�sthma [0, x] gia thn zx(t) =f(t)g(x)− g(t)f(x), brÐskoume ξ ∈ (0, x) ¸ste

0 = zx(x)− zx(0) = f ′(ξ)g(x)− g′(ξ)f(x).

AfoÔ h f ′

g′ eÐnai aÔxousa kai ξ < x, paÐrnoume

f(x)g(x)

=f ′(ξ)g′(ξ)

≤ f ′(x)g′(x)

.

Sunep¸c, f ′(x)g(x)− g′(x)f(x) ≥ 0. 'Epetai ìti h′ ≥ 0 sto (0, a), �ra h h eÐnaiaÔxousa.

G. Ask seic*

1. Melet ste thn g qwrist� sta diast mata (−∞, a1], [a1, a2], . . . , [an−1, an]kai [an, +∞) (gia na {di¸xete} tic apìlutec timèc). Ja qreiasteÐ na diakrÐnetetic peript¸seic n perittìc kai n �rtioc.

(a) An n = 2s−1 gia k�poion s ∈ N, elègxte ìti h g eÐnai fjÐnousa sto (−∞, as]kai aÔxousa sto [as,+∞). Sunep¸c,

min(g) =2s−1∑

k=1

|as − ak| =2s−1∑

k=s+1

ak −s−1∑

k=1

ak.

(b) An n = 2s gia k�poion s ∈ N, elègxte ìti h g eÐnai fjÐnousa sto (−∞, as],stajer sto [as, as+1], kai aÔxousa sto [as+1,+∞). Sunep¸c,

min(g) =2s∑

k=1

|as − ak| =2s∑

k=1

|as+1 − ak| =2s∑

k=s+1

ak −s∑

k=1

ak.

2. 'Estw n < m sto N. Efarmìzontac to je¸rhma mèshc tim c sto di�sthma[1m , 1

n

]brÐskoume ξnm ∈ (

1m , 1

n

)¸ste

∣∣∣∣f(

1n

)− f

(1m

)∣∣∣∣ = |f ′(ξnm)| ·∣∣∣∣1n− 1

m

∣∣∣∣ .

Apì thn upìjesh èqoume |f ′(ξnm)| ≤ 1, �ra

|am − an| ≤∣∣∣∣1n− 1

m

∣∣∣∣ ≤1n

.

Qrhsimopoi¸ntac to parap�nw deÐxte ìti h (an) eÐnai akoloujÐa Cauchy.

3. (a) Efarmìzontac to je¸rhma tou Rolle deÐxte epagwgik� to ex c: gia k�jek = 0, 1, . . . , n−1, h exÐswsh f (k)(x) = 0 èqei k diaforetikèc lÔseic sto di�sthma(−1, 1) kai f (k)(1) = f (k)(−1) = 0.(b) DeÐxte ìti h exÐswsh f (n)(x) = 0 èqei n diaforetikèc lÔseic sto di�sthma(−1, 1).(g) DeÐxte ìti h exÐswsh f (n)(x) = 0 èqei to polÔ n diaforetikèc lÔseic. [Upì-deixh: An h f (n)(x) = 0 eÐqe n + 1 diaforetikèc lÔseic, tìte h f (2n)(x) = 0 jaeÐqe lÔsh.]

4. 'Estw a > 0. Upojèste ìti up�rqei paragwgÐsimh sun�rthsh f : [0, 1] → Rme f ′(0) = 0 kai f ′(x) ≥ a gia k�je x ∈ (0, 1].(a) Qrhsimopoi¸ntac to je¸rhma mèshc tim c sto [0, x] deÐxte ìti gia k�je x ∈(0, 1) èqoume

f(x)− f(0)x

≥ a.

(b) Qrhsimopoi¸ntac thn f ′(0) = limx→0+

f(x)−f(0)x , mporeÐte na katal xete se

�topo.

5. QwrÐc periorismì thc genikìthtac, upojètoume ìti gia k�poio x0 ∈ (a, b)isqÔei

limx→x0

f ′(x) = ` > f ′(x0),

kai ja katal xoume se �topo.(a) JewroÔme m ∈ R o opoÐoc ikanopoieÐ thn ` > m > f ′(x0). DeÐxte ìtiup�rqei δ > 0 ¸ste (x0 − δ, x0 + δ) ⊂ (a, b) kai f ′(x) > m gia k�je x ∈(x0 − δ, x0) ∪ (x0, x0 + δ).(b) DeÐxte ìti h f ′ den èqei thn idiìthta Darboux sto (x0−δ, x0+δ) kai katal xteètsi se �topo.

6. Upojètoume ìti −∞ ≤ limx→b−

f ′(x) < +∞.

(a) DeÐxte ìti up�rqoun ` ∈ R kai x0 ∈ (a, b) ¸ste f ′(y) ≤ ` gia k�je y ∈ (x0, b).(b) Jewr ste x ∈ (x0, b) kai efarmìzontac to je¸rhma mèshc tim c sto [x0, x]deÐxte ìti

f(x) ≤ f(x0) + `(x− x0) ≤ f(x0) + `(b− a).

(g) Apì to (b) h f eÐnai �nw fragmènh sto [x0, b). Qrhsimopoi¸ntac thn upìjeshìti lim

x→b−f(x) = +∞ mporeÐte na katal xete se �topo.

7. Upojètoume ìti limx→+∞

f ′(x) = ` 6= 0.

(a) DeÐxte ìti up�rqei M > 0 ¸ste gia k�je x > M na èqoume |f ′(x)| > |`|2 .

(b) Jewr ste x > M kai efarmìzontac to je¸rhma mèshc tim c sto [M, x] deÐxteìti

|f(x)− f(M)| ≥ |`|(x−M)2

.

(g) AfoÔ limx→+∞

f(x) = L, èqoume limx→+∞

|f(x) − f(M)| = |L − f(M)|. Apì

thn �llh pleur�, limx→+∞

|`|(x−M)2 = +∞. Apì to (b) mporeÐte na katal xete se

�topo.(d) Upojètontac ìti lim

x→+∞f ′(x) = ±∞, mporeÐte p�li na deÐxete ìti up�rqei

M > 0 ¸ste gia k�je x > M na èqoume |f ′(x)| > 1. Sunep¸c, epanalamb�nontacta b mata (b) kai (g), katal gete se �topo.

Απειροστικός Λογισμός Ι - Σημειώσεις

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