I'm currently studying crystallographic groups and am reading through the paper An Account of the Theory of Crystallographic Groups by Louis Auslander. In the paper, there is a proof showing that a subgroup $\Gamma$ of the Euclidean motions $R(n)$ is discrete if and only if the sequence $\{\gamma_n\}$ in $\Gamma$ is eventually constant whenever the sequence $\{\gamma_n(x)\}$ is Cauchy, for any $x$ in the Euclidean space.
There is a line of reasoning in the proof that I can't quite jump to: Since $O(n)$ (orthogonal group) is compact we can find a subsequence of the $g_i$ which is Cauchy. Hence, the sequence $\gamma_i t_0$ is Cauchy. The part where we can find a subsequence that is Cauchy is fine to me, but I don't quite see how this implies that the sequence $\gamma_i t_0$ is Cauchy. I have attached a section of the proof below for reference.
To explain some of the notation as well, $E^n$ is defined as the Euclidean space in $n$ dimensions and $R^n$ is the subgroup of $R(n)$ containing the pure translations. Any explanation of this line of reasoning is much appreciated!
