# Alternate proof of a version of Arzela-Ascoli theorem

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?

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 $$x$$.