Corollary 2

Let $G$ be a commutative group and let $\mathfrak{T}_1, \mathfrak{T}_2$ be two topologies compatible with the group structure of $G$. Suppose that $\mathfrak{T}_1$ is finer than $\mathfrak{T}_2$ and that there is a fundamental system $\mathfrak{B}$ of neighbourhoods of $O$ with respect to $\mathfrak{T}_1$ which are complete with respect to the uniformity $U_2$ corresponding to $\mathfrak{T}_2$. Then $G$ is complete with respect to the uniformity $U_1$ corresponding to $\mathfrak{T}_1$.

The sets of $\mathfrak{B}$ are closed in the topology $\mathfrak{T}_2$, hence complete for the uniformity $U_1$ by Corollary 1; the result therefore follows from Proposition 4 of no. 3.