The question is the following: Let $f,f_1, f_2, ...$ be continuous real-valued functions on a compact metric space, $E$, where the $f_n$'s converge pointwise to $f$. We want to show that if $f_1(p) \leq f_2(p) \leq \dots$ for all $p \in E$ then the sequence of $f_n$'s converges uniformly.
I have been stuck on this problem for a while, and I cannot really get very far on it. I know that I can use Heine-Borel to say that $f, f_1, f_2\dots$ are all closed and bounded and that $f(p) = \sup_{n\in \mathbb{N}}\{ f_n(p)\}.$ But I'm not sure where to go from here.
Thanks for any help.