Corollary 3

Let $X$ be a locally compact space, $Y$ a Hausdorff uniform space, $H$ a subset of $C(X; Y)$. Then $H$ is relatively compact in $C_c(X; Y)$ if and only if $H$ is equicontinuous and $H(x)$ relatively compact in $Y$ for all $x \in X$.

In view of Corollary 1 it is enough to show that, if $H$ is relatively compact in $C_c(X; Y)$, then $H$ is equicontinuous. Now each point $x \in X$ has a compact neighbourhood $A$, and it follows from Theorem 2 that $H/A$ is equicontinuous; this implies that $H$ is equicontinuous at $x$, and the result is proved.