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Is the Arzelá-Ascoli Theorem true in a precompact subset of $\mathbb{R}^n$?. If $S \subset \mathbb{R}^n$ is precompact and we have a sequence $(f_n)$ of functions in $C(S)$ (Space of bounded and continuous functions $f: S \to \mathbb{R}$) which is equicontinuous and uniformly bounded, is there a subsequence of $(f_n)$ uniformly convergent?

I think that if $S=(a,b)$, with $a,b \in \mathbb{R}$, we can extend continuously each function $f_n$ to $[a,b]$ and apply Arzelá-Ascoli Theorem in $C([a,b])$.

Thanks in advance.

Mike Van
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    Consider $f_n(x) = x^n$ on $(0,1)$. You'd need uniform equicontinuity. If you have that, you can extend all functions continuously to $\overline{S}$ and the extended family is still equicontinuous. – Daniel Fischer Jan 27 '18 at 15:01
  • @DanielFischer So is the op's proposition true? I can't quite follow your comment: On one hand you state: "You'd need uniform equicontinuity." But you continue with "If you have that". With "that" do you mean equicontinuity? – ViktorStein Jul 24 '19 at 21:34
  • @ViktorGlombik The "that" in "If you have that" is uniform equicontinuity. With uniform equicontinuity each $f_n$ is uniformly continuous and thus has a continuous extension, call it $g_n$, to $\overline{S}$. The sequence of extensions is then also equicontinuous [and since $\overline{S}$ is compact, uniformly equicontinuous] and uniformly, hence pointwise, bounded. By Ascoli-Arzelà it has a uniformly convergent subsequence, and the corresponding subsequence of the $f_n$ converges uniformly on $S$. Just equicontinuity is not enough. The example in my first comment shows that. – Daniel Fischer Oct 02 '19 at 15:05
  • That the sequence is uniformly bounded is rather obvious. That it is equicontinuous follows easily from the locally uniform convergence of $f_n' \colon x \mapsto n\cdot x^{n-1}$ to $0$ on $S = (0,1)$, and that no subsequence converges uniformly on $S$ follows from the pointwise convergence to $0$ and $\lim_{x \to 1} f_n(x) = 1$ for all $n$. One point that muddies the waters is that many authors only consider compact domains when they introduce equicontinuity, and since there there is no difference between the concepts of equicontinuity and uniform equicontinuity omit the word "uniform". – Daniel Fischer Oct 02 '19 at 15:06

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