Definition:Cauchy Sequence in Topological Group

Definition
Let $G$ be a topological group with identity element $e$.

Let $(a_n)$ be a sequence in $G$.

Then $(a_n)$ is a Cauchy sequence for every neighborhood $U$ of $e$ there exists $N \in \N$ such that for all $m, n \geq N : a_na_m^{-1} \in U$.

Also see

 * Definition:Cauchy Sequence in Metric Space