Proposition 1

Tag 03N7, permanent. Cite it as [Bourbaki, Tag 03N7].

Theory of Sets, chapter III, § 6. Infinite sets

Every subset of a countable set is countable. The product of a finite family of countable sets is countable. The union of a sequence of countable sets is countable.

The first assertion is obvious. The others follow from the Corollaries to Theorem 2, no. 3.

¶ We have proved (no. 3, Lemma 1) that if $\mathfrak{a}$ is any infinite cardinal, then $\mathrm{Card}(\mathbf{N}) \leqslant \mathfrak{a}$ . This has the following consequences :

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

no. 3, Lemma 1
Theorem 2