Exercise 2, § 2

Let $X$ be a Hausdorff topological space in which every point admits a countable fundamental system of neighbourhoods, and let $Y$ be a uniform space. Let $H$ be a subset of $\mathscr{C}(X; Y)$ with the property that, for each compact subset $K$ of $X$, the set $H|K$ of restrictions to $K$ of mappings $u \in H$ is an equicontinuous subset of $\mathscr{C}(K; Y)$. Show that $H$ is equicontinuous (argue by contradiction).