Corollary 2
Tag 0BL0, permanent. Cite it as [Bourbaki, Tag 0BL0].
General Topology, chapter III, § 4. Groups operating properly on a topological space; compact ness in topological groups and spaces with operators
Let G be a Hausdorff topological group and let K be a compact subgroup of G. Then G is compact (resp. locally compact) if and only if G/K is compact (resp. locally compact).
Apply Corollary 1 to K operating on G by right translations.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.