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So there is this theorem of Arzela - Ascoli which is stated as below:

Given $X$, a compact metric space, $Y$ complete metric space, and let $\{f_n\}_{n\in \Bbb{N}} \subset C(X,Y)$, be an equicontinuous family of functions such that for every $x\in X$, $\{f_n(x)\}_{n \in \Bbb{N}}$ is relatively compact. Then there exist a subsequence of $(f_n)$ that converges uniformly.

My question is: Does this hold true for any general metric space Y or do we always need Y to be complete?

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Let $Z$ be the completion of $Y$. Then $(f_n)$ has a subsequence that converges uniformly to an element of $C(X,Z)$. However this limit must take values in $Y$ in view of the hypothesis that $(f_n(x))$ is relatively compact in $C(X,Y)$ for each $x$.

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