Wikipedia (https://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem#Compact_metric_spaces_and_compact_Hausdorff_spaces) asserts the following (specialized from Hausdorff spaces to metric spaces, for simplicity): if X,Y are metric spaces, where X is compact, then a pointwise precompact and equicontinuous subset F of C(X,Y) is precompact. I want to stress that it claims that completeness of Y is not necessary. Is that correct? If so, how does one prove that?
One approach seems to be to replace Y with its completion Z, so that any sequence in F has a subsequence which converges uniformly in C(X,Z), and then try to show that the limit is actually valued in Y. But I'm not sure how to go about doing so.
Thanks in advance!
Edit: I solved it: my approach does indeed work. The point is that once one has the limiting function valued in Z, then the pointwise precompactness implies that the limit is in Y.
I think you have the right idea. The fact that the family is assumed to be pointwise precompact seems like the hypothesis to use to say the limit function actually maps into $Y$ because this is a statement about precompactness in $Y$.