\begin{definition}[Definition:Cauchy Sequence in Topological Group]
Let $G$ be a topological group with identity element $e$.
Let $\sequence {a_n}$ be a sequence in $G$.
Then $\sequence {a_n}$ is a '''Cauchy sequence''' {{iff}} for every neighborhood $U$ of $e$ there exists $N \in \N$ such that for all $m, n \ge N : a_n a_m^{-1} \in U$.
\end{definition}