Suppose $G$ is a topological group of homeomorphisms of a topological space $X$ (which is Hausdorff and countable at infinity), such that $G$ acts transitively, and has got $Stab_G(x)$ compact for each $x\in X$.
Then, if $\Gamma$ is a discrete subgroup of $G$, then $\Gamma$ acts properly (which here means that for every compact set $K\subseteq X$, $\sharp\{g\in\Gamma\,|\,K\cap gK\not=\emptyset\}<+\infty$).
Can someone show me how to prove this, and pinpoint whether the assumption made on $G$ are necessary?