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