3. If g:A → R is continuous and (an) is a Cauchy sequence in A, does it follow that the sequence (g(an)) is Cauchy?

6. Assume f: R → R and for some c < 1 we have for all x, y € R, |f(x) = f(y)| ≤ c|x − y\. (a) Prove that f is uniformly continuous. (b) Pick and ₁ R and inductively define a sequence by letting In = f(n-1) (for n ≥ 2). Prove that the sequence (n) is Cauchy. (c) If (x) is the sequence from (b), and (xn) → a, then f(a) = a (so a is a fixed point of f). (d) Prove that a the only fixed point of f.

5. If an is absolutely convergent, is it true that every rearrangement of an is also absolutely convergent? ∞

1. Construct a continuous function f : X → Y between two metric spaces with the condition that there exists a Cauchy sequence {n}₁ in X but {f(xn)}~_=₁ is not a Cauchy sequence in Y. n=1