Proposition 9

Tag 0C2W, permanent. Cite it as [Bourbaki, Tag 0C2W].

General Topology, chapter IX, § 2. Metric spaces and metrizable spaces

A metric space $X$ is complete if and only if every Cauchy sequence in $X$ is convergent.

Let $\hat{X}$ be the completion of $X$ . If there is a point $x \in \hat{X}$ which does not belong to $X$ , then there is a sequence $(x_n)$ of points of $X$ which converges to $x$ , and this is a non-convergent Cauchy sequence in $X$ .

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

Cited by

Proposition 5, § 2
Proposition 15, § 2
Proposition 4, § 3

Cites

Nothing.