I am a bit confused because the classical Arzela-Ascoli gives uniform convergence of a sequence (taking a sub-sequence if necessary) of functions. However the more general results seem to require pointwise convergence first.
Given a compact Hausdorff metric space $X$ and a complete metric space $Y$, and sequence of continuous functions $f_n:X\to Y$. I wonder if anyone has a reference for Arzela-Ascoli in this particular case to show that $f_n$ converges uniformly to some $f$ in the metric of $Y$ (taking sub-sequence if necessary). Where we don't a-priori have pointwise convergence.
Note : Proposition 3.3.1 of the book (Gradient Flows in Metric Spaces by Ambrosio, picture below) provides a Arzela-Ascoli pointwise convergence result, it seems like combining this with the results found on the wiki https://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem#Generalizations we get uniform convergence. I'm finding it hard to reconcile Proposition 3.3.1 with the wikipedia. 