Corollary

In the conditions of Proposition 7, if $U_2$ is a uniformity of a complete space, then so is $U_1$.

For every Cauchy filter with respect to $U_1$ is then a Cauchy filter with respect to $U_2$ and therefore converges in the topology $\mathfrak{T}_2$.

Note that the hypotheses of the Corollary to Proposition 7 are satisfied when $\mathfrak{T}_1 = \mathfrak{T}_2$ (§ 1, no. 2, Corollary 2 to Proposition 2).