Suppose $H_n,\ n\in\mathbb{N}$, and $H_0$ are subsets of a semimetric space $H$ such that
- every $h\in H_0$ is a limit of a sequence $h_n\in H_n$,
- if a subsequence $h_{n_j}$ converges to a limit $h$, then $h\in H_0$.
Suppose that $x$, $x_n,\ n\in\mathbb{N}$, are real bounded functions defined on $H$, such that $x_n$ converges to $x$ uniformly and $x$ is uniformly continuous. I need to show that $S_n:=\sup_{h\in H_n}x_n(h)\to\sup_{h\in H_0}x(h)=:S_0,\ n\to\infty.$
I succeeded in showing that for an arbitrary $\varepsilon$ and $n$ large enough $S_0
P.S. $H$ can be assumed to be totally bounded.