Corollary

Tag 0BR8, permanent. Cite it as [Bourbaki, Tag 0BR8].

General Topology, chapter IV, § 5. Real-valued functions

An increasing (resp. decreasing) real-valued function, defined on a directed subset $A$ of an ordered set $X$ , has a finite limit with respect to $A$ if and only if it is bounded above (resp. bounded below) in $A$ .

If we apply Theorem 2 to the case where $A = X = \mathbf{N}$ (ordered by the relation $\leq$ ), we have the following proposition:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Theorem 1, § 7

Cites

Theorem 2