Corollary
Tag 05DV, permanent. Cite it as [Bourbaki, Tag 05DV].
Algebra, chapter I, § 6. Extensions, solvable groups, nilpotent groups
Every p-group is nilpotent.
This follows from no. 3, Proposition 7.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.