[math] \newcommand{\mathds}{\mathbb}[/math]

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Given a matrix [math]M\in M_N(\mathbb C)[/math] having norm [math]||M||\leq1[/math], prove that

[[math]] P=\lim_{n\to\infty}\sum_{k=1}^nM^k [[/math]]

exists, and equals the projection onto the [math]1[/math]-eigenspace of [math]M[/math].