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I was going through some old notes and I came across a statement as follows. Suppose $E$ is a compact metric space and $K$ is a compact subspace of a normed space $X$. Then if we have an equicontinuous sequence of maps $f_n$ from $E$ to $K$, then it has a uniformly convergent subsequence. Maybe I'm missing something, but does this version of Arzela-Ascoli hold without any boundedness condition? I should think not, but maybe I'm wrong. I'd like to know of a counterexample or an idea of proof. Thanks.

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There is a boundedness condition: namely the condition that $K$ is compact. Indeed, this is quite a bit stronger than saying that the sequence is pointwise bounded: it saying there is a single compact (not just bounded) set that contains the entire image of every $f_n$.

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