## Question

2. Suppose K is compact, r > 0 and E = { f ∈ G(K) : |f(x)| ≤ r, all x ∈ K }.

Show that E is closed in G(K).

3. Suppose, f, f₁, f₂, .... are functions from a set E to a metric space (X, d), where E is a metric space with metric d. Show ժ(f(xₙ), f(yₙ)) --> 0 whenever {xₙ}, {yₙ} are sequences in E with d(xₙ, yₙ) --> 0. Hint: In one direction, prove by contradiction.

4. a) Suppose f: X --> Y, X, Y metric spaces and that f is uniformly continuous. Show that {f(xₙ)} is a Cauchy sequence in Y whenever {xₙ} is a Cauchy sequence in X.

b) Use the example f(x) = x² (X = Y = R) to show the converse of the statement in a) is false. Hint: Cauchy => Bounded.

5. a) Suppose f: [0, ∞) --> R and that f is uniformly continuous on both [0, 1] and [1, ∞). Show that f is uniformly continuous on [0, ∞).

Hint: Make sure to consider the case 0 < x < 1 < u.

b) Define f: [0, ∞) --> R by f(x) = √x. Show that f is uniformly continuous.

## Solution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

By purchasing this solution you'll be able to access the following files:

Solution.pdf.