Let X be a compact metric space and Y be a normed space.
Prove that if f_n belongs to C(X,Y), then lim_n f_n = f_o in the Sup norm
if and only if lim_n f_n = f_o uniformly in X.
[ Note: Sup norm: ||f|| = Sup||f(x)|| for every x in X.]
If f_n->f_o as n->oo in the Sup norm, then for any e>0, there exists some integer N>0, such that for all n>N, we have ||f_n(x)-f_o(x)||=Sup|f_n(x)-f_o(x)|<e. Then for any x ...
This is a proof regarding a compact metric space and a normed space.