Equivalence to properly discontinuous action

Equivalence to properly discontinuous action

Let $X$ be a metric space and let $G$ be a group of homeomorphisms $X \to
X$ acting on $X$. We say $G$'s action is properly discontinuous in case
for every $x \in X$ and compact $K \subseteq X$, there are at most
finitely many $g \in G$ such that $g(x) \in K$. Equivalently (and this is
not hard to show), $G \cdot x$ is discrete and $G_x$ finite for any $x$.
Why is it the case that $G$ acts properly discontinuously if and only if
for any compact $K$, $g(K) \cap K \ne \emptyset$ for only finitely many
$g$? One direction is relatively easy, but I just cannot seem to prove the
"only if". The best I've been able to do is for finite $K$ (which is kind
of the next best thing when you're stumped on proving something for
compact sets, I guess).