Let $f_n : [-2,2] \to [0,1]$ be a sequence of convex functions. Show that there is a sub-sequence ${f_{n_k}}$ that converges uniformly on $[-1,1]$.
When I see this problem, I immediately think Arzela-Ascoli because it asks to show that a sequence of functions has a uniformly convergent sub-sequence. It gives us that they are uniformly bounded, but for some reason I can't convince myself (or prove) that a family of bounded convex functions on a compact interval must be equicontinuous. Perhaps this is the wrong approach and there is something more immediate. Any suggestions? Thanks in advance.