Problem Set 2

Due date: November 8, in class

Exercise 1. Show that if a and b are both limits of the sequence {xn}, then a = b.

Exercise 2. Define the sequence {xk}∞k=1 with xk =∑k

i=11i2

. Show that the sequence converges.

Recall that a function f : (X, d)→ (Y, ρ) is continuous on A ⊂ X if for all ε > 0 and x0 ∈ A, thereexists δ(x0, ε) > 0 such that ρ (f(x0), f(x)) < ε for all x ∈ A, d(x, x0) < δ(x0, ε). Note that thenotation δ(x0, ε) means that the choice of δ may depend on x0 and ε. The function is uniformlycontinuous on A if the δ does not depend on x0.

Exercise 3. Show that the function f(x) = x2 is continuous but not uniformly continuous on(0,∞).

Exercise 4. Consider a sequence of functions fn : [0, 1]→ R, defined by fn(x) = xn for x ∈ [0, 1].{fn} converges pointwise to function f if for any x ∈ [0, 1], limn→∞ fn(x) = f(x).

1. Show that {fn} is pointwise convergent and find its limit f .

2. Show that fn is continuous but f is not.

3. The sup norm for a real valued function g defined on a set X is ||g||s = sup{|g(x)|, x ∈ X}.Show that {fn} does not converge to f in the sup norm.

Exercise 5. Define the function f(x1, x2) = x41x2 − x21x32. Calculate ∇f(x) and find the points xwhere ∇f(x) = (0, 0).

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