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Let ##X## be a compact metric space. Prove that if ##\mathcal F \subset X## is a family of equicontinuous functions ##f:X \to Y \implies \mathcal F## is uniformly equicontinuous.

The attempt at a solution.

What I want to prove is that given ##\epsilon>0## there exists ##\delta>0##: if ##d_X(x,y)<\delta \implies \forall f \in \mathcal F d_Y(f(x),f(y))<\epsilon##. I know that for an arbitrary ##x_0 \in X## and a given ##\epsilon## I can find ##\delta##. I also know that ##X## is compact. I think I should write ##X## as a union of open covers involving something with the ##\delta## that works for each point ##x_0##, then extract a finite subcover (I would have finite ##\delta##'s) and take the minimum of those deltas. I got stuck trying to find the proper union of open covers.