Lemma 1
Tag 03DQ, permanent. Cite it as [Bourbaki, Tag 03DQ].
Théories spectrales, chapter V, § 4. Représentations des groupes compacts
Every irreducible unitary representation of G is square-integrable.
In fact, the matrix coefficients of an irreducible unitary representation are continuous and bounded, and are therefore square-integrable on G, since G is compact.
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.
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