Proposition 4

If, in a topological group $G$, there is a neighbourhood $V$ of $e$ which is complete with respect to either the right or the left uniformity, then $G$ is complete.

Suppose for example that $V$ is complete with respect to the right uniformity, and let $\mathfrak{F}$ be a Cauchy filter on $G_d$; then $\mathfrak{F}$ contains a $V_d$-small set $M$, and if $x_1 \in M$ we therefore have $M \subset Vx_1$. Hence the trace of $\mathfrak{F}$ on the complete subspace $Vx_1$ of $G_d$ is a Cauchy filter, which converges to a point $x_0$; since $x_0$ is a cluster point of $\mathfrak{F}$, it is a limit of $\mathfrak{F}$ (Chapter II, § 3, no. 2, Corollary 2 to Proposition 5).