Remark 5

Tag 024X, permanent. Cite it as [Bourbaki, Tag 024X].

Théories spectrales, chapter I, § 1. Spectres et caractères

Suppose that the algebra A is nonzero. Let $x\in A$ be a nilpotent element. Let $n$ be an integer such that $x^n= 0$ . The spectrum of $x^n$ reduces to 0, hence the same is true of the spectrum of $x$ by Remark 4.

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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