Corollary 1

Let $(X_\alpha, f_{\alpha \beta})$ be an inverse system of sets indexed by a directed set $I$ which has a countable cofinal subset, and suppose that the $f_{\alpha \beta}$ are surjective. Then if $X = \varprojlim X_\alpha$, the canonical mapping $f_\alpha : X \to X_\alpha$ is surjective for each $\alpha \in I$.

Let each $X_\alpha$ carry the discrete uniformity, and apply Theorem 1.