Corollary

— Let $\mathfrak{F}$ be a set of numerical functions $\geq 0$, directed for the relation $\leq$, such that there exists a countable cofinal subset $\mathfrak{G}$ of $\mathfrak{F}$ (S, III, §1, No. 7); then

For, there exists an increasing sequence of functions in $\mathfrak{G}$ having the same upper envelope as $\mathfrak{F}$: if $(f_n)$ is the sequence of functions of $\mathfrak{G}$, arranged in any order, let $(f_{n_k})$ be a subsequence defined recursively by the conditions $n_1 = 1$, $f_{n_{k+1}} \geq \sup(f_{n_k}, f_k)$; it is clear that this subsequence has the indicated properties.