Proposition 1

Let G be a topological group operating continuously on a topological space X, and let K be a quasi-compact subset of G. Then the mapping $\rho : (s, x) \to s.x$ of $K \times X$ into X is proper.

$\rho$ factorizes into $K \times X \to K \times X \xrightarrow{\mathrm{pr}_2} X$, where $\alpha(s, x) = (s, s.x)$. $\alpha$ is a homeomorphism, for $\alpha^{-1} : (s, y) \to (s, s^{-1}.y)$ is continuous. Since K is quasi-compact, $\mathrm{pr}_2$ is proper (Chapter I, § 10, no. 2, Theorem 1, Corollary 5); hence $\rho$ is proper (Chapter I, § 10, no. 1, Proposition 5).