Theorem 2 (Theorem of the monotone limit)

Let $X$ be an ordered set and let $A$ be a directed subset of $X$ (*). Every monotonic real-valued function $f$ defined on $A$ has a limit with respect to $A$ (Chapter I, , no. 3); if $f$ is increasing (resp. decreasing), this limit is equal to the least upper bound (resp. greatest lower bound) of the set $f(A) \subset \overline{\mathbf{R}}$

Suppose for example that $f$ is increasing, and let $a = \sup f(A)$. If $a = -\infty$, the theorem is trivial. If $a > -\infty$, then for each $b < a$ there exists $x \in A$ such that $b < f(x) \leq a$; hence, if $S_x$ is the section of $A$ relative to $x$ (i.e. the set of all $y \geq x$, cf. Chapter I, § 6, no. 3), $f(S_x)$ is contained in the neighbourhood $]b, +\infty]$ of $a$, and the theorem follows. The proof is analogous if $f$ is decreasing.