I need some help with this one.

A sequence of functions f_n is uniformly bounded on a set E iff there is an M>0 s.t. abs(f_n(x)) <= M for all x in E and n in N. Suppose each f_n is a bounded function and f_n converges to F uniformly on E.

Prove {f_n}is uniformly bounded on E and f is a bounded function on E.

Proof:

Since f_n is bounded we know that there exists m<= f_n <=M and we know since f_n converges to f uniformly then for all ε>0 there exists an N inN,s.t. n ≥ N implies (f_n(x)-f(x)) < ε for all of x in E. Then f_n converges and is therefore bounded.

Since f_n <= M then f_n is uniformly bounded.

I'm not sure where to go from here.