$A_n(s)$ is a sequence of convex random functions defined on an open set $S\in \mathbb{R}^p$ which converges in probability to some $A(s)$ for each $s$. I'm trying to show that $\sup_{s\in K} \big| A_n(s)-A(s) \big|\stackrel{P}{\to} 0$ for $K\subset S$ compact.
I think I could use the fact that since $K$ is compact, then $\sup_{s\in K} \big| A_n(s)-A(s) \big| > \epsilon$ is an event contained in $\bigcup_{s\in K} \left\{\big| A_n(s)-A(s) \big| > \epsilon\right\}$ (since continuous functions reach their supremums on compact sets). But after that, I'm stuck because I can't see how to show the probability of $\bigcup_{s\in K} \left\{\big| A_n(s)-A(s) \big| > \epsilon\right\}$ goes to $0$.
Suggestions?