Spectral radius/Maximum norm/Fact/Proof

Let ${\displaystyle {}\lambda }$ be an eigenvalue of ${\displaystyle {}\varphi }$ with

${\displaystyle {}\vert {\lambda }\vert =\rho (\varphi )\,.}$

Let ${\displaystyle {}v\in V}$ be an eigenvector of ${\displaystyle {}\lambda }$. Then

${\displaystyle {}\vert {\lambda }\vert \cdot \Vert {v}\Vert =\Vert {\lambda v}\Vert =\Vert {\varphi (v)}\Vert \leq \Vert {\varphi }\Vert _{\rm {max}}\cdot \Vert {v}\Vert \,,}$

and division by ${\displaystyle {}\Vert {v}\Vert }$ yields the claim.