Corollary 4

Let $X$ be a Hausdorff space, and let $f_i : X \to Y_i$ be a family of proper mappings. Then the mapping $f : x \to (f_i(x))$ of $X$ into $\prod_i Y_i$ is proper.

The proof is the same as in the case of a finite family (no. 1, Proposition 5, Corollary 3), using Corollary 3 above and the fact that the diagonal of $X^1$ is closed (§ 8, no. 1, Proposition 1).