Let $K$ be a compact metric space and $S\subset C(K,\mathbb{C})$.
Let $S$ be closed,bounded and equicontinuous. The usual proof for this is, using Arzela-Ascoli Theorem and Axiom of countable choice, showing that $S$ is limit point compact. (The problem is, under ZF, additional hypothesis "$K$ is separable" should be added to make Arzela-Ascoli Theorem true)
I think, in order to prove $S$ is compact without AC, one should start directly with an infinite subcover.
Assuming above hypotheses, I have proved $S$ is totally bounded and complete, and it really does seem provable that $S$ is compact under ZF.
Is it unprovable? Or if there is an argument proving this without choice please let me know help!
Thank you in advance