Endomorphism/Power sequence/Convergence/Eigenvector or 0/Fact

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, and let

$\varphi \colon V\longrightarrow V$

be an endomorphism. Let ${}v\in V$ be such that the sequence ${}\varphi ^{n}(v)$ converges.

Then the

limit vector

${}w:=\lim _{n\rightarrow \infty }\varphi ^{n}(v)\,$

is the zero vector, or an eigenvector of ${}\varphi$ for the eigenvalue

${}1$.

Proof, Write another proof