$G$ is called $n$-dimensional crystallographic group if it's a discrete subgroup of $\operatorname{Isom}(\mathbb{R}^n)$ acting on $\mathbb{R}^n$ with compact fundamental domain.
An action of a topological group $G$ on a topological space $X$ is called proper if the map $\phi:G\times X \rightarrow X\times X$ with $\phi(g,x) = (x,gx)$ is a proper map.
I should prove that the action of a $n$-dimensional crystallographic group on $\mathbb{R}^n$ is proper, but I'm missing a clever way. How can I prove it?
