Let $\{f_n\}$ be a uniformly bounded sequence of continuous functions such that $f_n(x) \leq f_{n+1}(x)$ for all $x \in X$ where $X$ is compact. Prove $\{f_n\}$ converges uniformly to some function $f: X \to \mathbb{R}$.
For starters, we can use monotonicity and the uniformly boundedness to show point-wise convergence. By Dini's theorem, the convergence is uniform if and only if $f$ is continuous. I'm having some trouble showing that the convergence is uniform or that $f$ is continuous.