Complex Analysis Problems

1. Suppose that fₙ: [0, 1] --> R has the graph clearly fₙ --> 0 on [0, 1] (see the attached sketch). Show that fₙ ⇏ 0 [0, 1]. Hint: ε = 1.

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.

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