Exercise 7, § 4

Let $G$ be a Hausdorff group operating continuously on a locally compact space $X$, and suppose that (i) for every compact subset $K$ of $X$, the restriction to $G \times K$ of the mapping $(s, x) \to s.x$ is a closed mapping of $G \times K$ into $X$; (ii) for each $x \in X$, $s \to s.x$ is a proper mapping of $G$ into $X$. Show that $G$ operates properly on $X$ (use Proposition 12).