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517 HW5 1. Let (X, dX ) and (Y, dY ) be metric spaces with X compact, let Z = {f : X → Y | f is continuous}, and define dZ : Z × Z → R by dZ (f, g) = sup dY (f (x), g(x)). x∈X Prove that dZ is a metric. 2. Let Z, X and Y be as in Problem 1, and consider continuous functions fn : X → Y . Prove that fn converges uniformly if and only if {fn } converges in (Z, dZ ). 3. Give an example of sequences {fn }, {gn } of uniformly converging functions such that {fn gn } does not converge uniformly. 4. Define fn (x) = x . 1 + nx2 Prove that {fn } converges uniformly to a function f . Show that f 0 (x) = limn→∞ fn0 (x) except when x = 0. 5. Prove that every uniformly convergent sequence of bounded functions is uniformly bounded. 1