Exercise 35, § 2

Let U be a non-empty open convex set in $\mathbf{R}^n$ and $\mathscr{F}$ be a set of convex functions defined over U. Let $\Phi$ be a filter on $\mathscr{F}$ that converges pointwise in U to a finite function $f_0$; show that $\Phi$ converges uniformly to $f_0$ in every compact subset of U (use exerc. 34).

EXERCISES