Crystallographic group property

Question

I'm studying on the article "A Geometric proof of Bieberbach's theorems on crystallographic groups".

G is called n-dimensional crystallographic group if it's a discrete subgroup of $Isom(\mathbb{R}^n)$ acting on $\mathbb{R}^n$ with compact fundamental domain.

I don't understand the first statement of this proposition. I'm assuming it's a consequence of the discreteness, but I can't find a way to prove it.

Answer 1

Note that any bounded infinite set has a limit point therefore cannot be discrete. In particular the hypersphere of radius $t$ centered at $a$ is bounded so it may only intersect with finitely many cosets or $G$ would have a limit point.