Corollary 1
Tag 0BK3, permanent. Cite it as [Bourbaki, Tag 0BK3].
General Topology, chapter III, § 3. Uniform structures on groups
A locally compact group is complete.
For every compact space is complete with respect to its unique uniformity (Chapter II, § 4, no. 1, Theorem 1).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.