I was actually thinking that I could show that (f_n - f)g is continuous, on the interval for all n since f is bounded on [a,b] and f_n is bounded for all n on [a,b]. That still doesn't use the endpoints though, which I'm figuring are key. I do understand what you are saying though, I was trying...
Homework Statement
Let \left[a,b\right] be a closed bounded interval, f : [a,b] \rightarrow \textbf{R} be bounded, and let g : [a,b] \rightarrow \textbf{R} be continuous with g\left(a\right)=g\left(b\right)=0. Let f_{n} be a uniformly bounded sequence of functions on \left[a,b\right]. Prove...