Exercise 1, § 4

a) Let $H$ be a set of convex functions on a compact interval $[a, b] \subset \mathbf{R}$; suppose that the sets $H(a)$ and $H(b)$ are bounded above in $\mathbf{R}$ and that there exists a point $c$ such that $a < c < b$ and that $H(c)$ is bounded below in $\mathbf{R}$; show that $H$ is an equicontinuous set on $]a, b[$ (Gen. Top., X, p. 283)

b) Let $H$ be a set of convex functions on an interval $I \subset \mathbf{R}$, and let $\tilde{\mathscr{F}}$ be a filter on $H$ which converges pointwise on $I$ to a function $f_0$; show that $\tilde{\mathscr{F}}$ converges uniformly to $f_0$ on every compact interval contained in $I$.