Corollary 2
Tag 0C2P, permanent. Cite it as [Bourbaki, Tag 0C2P].
General Topology, chapter IX, § 2. Metric spaces and metrizable spaces
Every countable product of metrizable uniform spaces is metrizable.
For such a space is Hausdorff and its uniformity has a countable fundamental system of entourages (Chapter II, § 2, no. 6).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.