Proposition 5

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

General Topology, chapter IV, § 7. Infinite sums and products of real numbers

An infinite product of finite real numbers is commutatively convergent if and only if it is absolutely convergent.

This follows from Chapter III, § 5, no. 7, Proposition 9, and from Theorem 4 above.

Moreover, the product whose general factor is $1 + u_n$ is absolutely convergent if and only if the series whose general term is $u_n$ is absolutely convergent.

A product of non-zero real numbers can be convergent without being commutatively convergent, i.e. without being absolutely convergent.

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

Cited by

Nothing.

Cites

Chapter III, § 5, no. 7, Proposition 9
Theorem 4