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Suppose $H_n,\ n\in\mathbb{N}$, and $H_0$ are subsets of a semimetric space $H$ such that

  1. every $h\in H_0$ is a limit of a sequence $h_n\in H_n$,
  2. 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 $S_0$x$. 2. suggests that I should look for a convergent subsequence of $h_n$, but as far as I can see that would result in a particular subsequence of $S_n$ converging to $S_0$. And I don't see how to find such a subsequence.

P.S. $H$ can be assumed to be totally bounded.

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