Corollary 2

Let X, L be two sets filtered by filters $\mathfrak{T}, \mathfrak{G}$ respectively, and let Y be a complete uniform space. For each $\lambda \in L$, let $u_\lambda$ be a mapping of X into Y. Suppose that (i) the family $(u_\lambda)_{\lambda \in L}$ converges uniformly in X (with respect to the filter $\mathfrak{G}$) to a mapping $v : X \to Y$; (ii) for each $\lambda \in L$, $u_\lambda$ has a limit $y_\lambda$ with respect to the filter $\mathfrak{T}$. Under these conditions, v has a limit with respect to $\mathfrak{T}$, and every limit of v with respect to $\mathfrak{T}$ is a limit of the family $(y_\lambda)_{\lambda \in L}$ with respect to $\mathfrak{G}$.

For v lies in the closure of the set of the $u_\lambda$ in $F_u(X; Y)$, and therefore $v(\mathfrak{T})$ is a Cauchy filter base on Y by virtue of Proposition 8; this shows that v has a limit y with respect to $\mathfrak{T}$ because Y is complete. Let

X' = X \cup \{ \omega \} be the topological space associated with the filter $\mathscr{F}$ (Chapter I, § 6, no. 5), and extend $u_\lambda$ (resp. $v$) to a mapping $\bar{u}_\lambda$ (resp. $\bar{v}$) of $X'$ into $Y$ by putting $\bar{u}_\lambda(\omega) = y_\lambda$ [resp. $\bar{v}(\omega) = y$]. Then the mappings $\bar{u}_\lambda, \bar{v}$ are continuous on $X'$, and $\bar{u}_\lambda$ converges uniformly in $X$ to $\bar{v}$ with respect to $\mathscr{G}$; since $X$ is dense in $X'$, the Corollary to Proposition 6 shows that $\bar{u}_\lambda$ converges uniformly in $X'$ to $\bar{v}$, and in particular that $y = \lim_{\mathscr{G}} y_\lambda$.