Let $F \subset \mathcal{C}([0, 1], \mathbb{R})$ be closed, equicontinuous, and pointwise bounded. Let $I : F → \mathbb{R}$ be defined by $I(f) = \int\limits_0^1 f(x)dx$. Show that there is $f_0 \in F$ such that $I(f_0) ≥ I(f)$ for all $f \in F$.
By Ascoli's theorem, $F$ is clearly compact. Do I then need to show that there is a maximal element?