Nilpotent mappings/Commute/Sum/Exercise

Let ${}V$ be a ${}K$-vector space, and let

$\varphi ,\psi \colon V\longrightarrow V$

denote nilpotent linear mappings, satisfying the relation

${}\varphi \circ \psi =\psi \circ \varphi \,.$

Show that also ${}\psi +\varphi$ is nilpotent.

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