Corollary 2

Let $\mathscr{S}_1, \mathscr{S}_2$ be two sets of subsets of $X$, whose union is the same and which are such that $\mathscr{S}_1 \subset \mathscr{S}_2$, and let $H$ be a subset of $\mathscr{F}(X; Y)$. Then if $H$ is complete with respect to $\mathscr{S}_1$-convergence, it is complete with respect to $\mathscr{S}_2$-convergence.

For every Cauchy filter with respect to $\mathscr{G}_2$-convergence is also a Cauchy filter with respect to $\mathscr{G}_1$-convergence, and we may apply Corollary 1.