Proposition 10

*In the product space $\overline{\mathbf{R}}^X$ the upper envelope $\sup_{i \in I} f_i$ of a family of real-valued functions $(f_i)_{i \in I}$ is the limit, with respect to the directed set $\mathfrak{F}(I)$ of finite subsets of $I$, of the mapping $H \to \sup_{i \in H} f_i$ [which maps each finite subset $H$ of $I$ to the upper envelope of the finite subfamily $(f_i)_{i \in H}$].

This follows immediately from Proposition 5 of no. 4 and from Chapter I, § 7, no. 6, Corollary 1 to Proposition 10.

We may therefore write