Endomorphism/Nilpotent vector/Invariant subspace/Exercise

Let ${}\varphi \colon V\rightarrow V$ a linear mapping on a ${}K$-vector space ${}V$ over a field ${}K$. Show that the subset of ${}V$, defined by

${}U={\left\{v\in V\mid {\text{ there exists an }}n\in \mathbb {N} {\text{ with }}\varphi ^{n}(v)=0\right\}}\,,$

is an

${}\varphi$-invariant linear subspace.

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