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I am given a corollary of the Arzela-Ascoli theorem, and I've substantially rephrased it to this:

If $S$ is an equicontinuous and pointwise bounded set of functions with domain a compact metric space and codomain $\mathbb R^m$, and if $f_k$ is a sequence in $S$, then $f_k$ has a uniformly convergent subsequence.

1) Is the above statement fully correct?

2) Does the uniform limit of the subsequence (in the conclusion of the statement) necessarily lie in $S$ itself?

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1) Yes, the statement is correct.

2) $S$ is not necessarily closed: take $(f_n,n\geqslant 1)$ a sequence of continuous functions which converges uniformly to $f$. Then $(f_n,n\geqslant 1)$ is equicontinuous and pointwise bounded, but $S:=\{f_n,n\geqslant 1\}$ is not closed unless $f=f_n$ for some $n$.

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  • $\begingroup$ Thanks! For my second question: does the uniform limit mentioned in the conclusion of the statement necessarily lie in S itself? $\endgroup$
    – ryang
    Commented Sep 29, 2013 at 17:12
  • $\begingroup$ Not necessarily (for example $f_n=1/n\cdot g$, where $g$ is continuous. $\endgroup$ Commented Sep 29, 2013 at 17:13

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