In my differential geometry course, we saw that a group of diffeomorphism $G$ of a smooth manifold $M$ acts properly discontiuously on $M$ if
- each $x \in M$ admits an open neighborhood $U$ such that $\varphi(U) \cap U = \varnothing$ for all $\varphi \in G\backslash \{\text{id}\}$,
- for any pair $x, y \in M$ such that $x$ is not in the orbit of $y$ under $G$, there exists two neighborhoods $U$ and $V$ of $x$ and $y$ respectively such that $\varphi(U) \cap \psi(V) = \varnothing$ for all $\varphi, \psi \in G$.
And then the professor add a remark saying that if a finite group acts freely on $M$, then it also acts properly discontinuously. I really have no idea where to start, does anyone have a hint? Thank you!