Suppose that a smooth manifold $M$ is a metric space and that $\Gamma$ is a discrete group of smooth isometries acting discontinuously on $M$. Show that the action is necessarily properly discontinuous.
Discontinuous means: each $x \in M$ has a neighborhood $U$ such that $\{h \in \Gamma ; hU \cap U \neq \emptyset\}$ is finite.
And properly discontinuous means both discontinuous and if $x,y \in M$ are not in the same orbit, then there are neighborhoods $U, V$ of $x,y$ such that $U \cap \Gamma V = \emptyset$.
Anyone can solve this problem? Thanks.