Advanced Calculus (I)

WEN-CHING LIEN

Department of MathematicsNational Cheng Kung University

WEN-CHING LIEN Advanced Calculus (I)

2.1 Limits Of Sequences

DefinitionA sequence of real numbers {xn} is said to converge to areal number a ∈ R if and only if for every ε > 0 there is anN ∈ N (which in general depends on ε) such that

n ≥ N implies |xn − a| < ε

2.1 Limits Of Sequences

DefinitionA sequence of real numbers {xn} is said to converge to areal number a ∈ R if and only if for every ε > 0 there is anN ∈ N (which in general depends on ε) such that

n ≥ N implies |xn − a| < ε

Example:

Prove that1n→ 0 as n→∞.

Example:

Prove that1n→ 0 as n→∞.

Example:

The sequence {(−1)n}n∈N has no limit.

Example:

The sequence {(−1)n}n∈N has no limit.

Remark:

A sequence can have at most one limit.

Remark:

A sequence can have at most one limit.

DefinitionBy a subsequence of a sequence {xn}n∈N, we shall meana sequence of the form {xnk}k∈N, where each nk ∈ N andn1 < n2 < . . .

Remark:

If {xn}n∈N converges to a and {xnk}k∈N is anysubsequence of {xn}n∈N, then xnk converges to a ask →∞.

Remark:

If {xn}n∈N converges to a and {xnk}k∈N is anysubsequence of {xn}n∈N, then xnk converges to a ask →∞.

DefinitionLet {xn} be a sequence of real numbers.

(i){xn} is said to be bounded above if and only if there isan M ∈ R such that xn ≤ M for all n ∈ N

(ii) {xn} is said to be bounded below if and only if there isan m ∈ R such that xn ≥ m for all n ∈ N

(iii) {xn} is said to be bounded if and only if it is boundedboth above and below.

TheoremEvery convergent sequence is bounded.

Exercise:

1 limn→∞

5 + nn2

2 Suppose that limn→∞

xn = 1

Find limn→∞

2 + xn2

Exercise:

1 limn→∞

5 + nn2

xn = 1

Find limn→∞

2 + xn2

Exercise:

1 limn→∞

5 + nn2

xn = 1

Find limn→∞

2 + xn2

Thank you.

Advanced Calculus (I)

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