Endomorphism/K/Powers/Zero convergence/Fact

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

$\varphi \colon V\longrightarrow V$

be an endomorphism. Then the following properties are equivalent.

1. ${}\varphi$ is asymptotically stable.
2. For every ${}v\in V$, the sequence ${}\varphi ^{n}(v)$, ${}n\in \mathbb {N}$, converges to ${}0\in V$.
3. There exists a generating system ${}v_{1},\ldots ,v_{m}\in V$ such that ${}\varphi ^{n}(v_{j})$, ${}j=1,\ldots ,m$, converges to ${}0$.
4. The modulus of every complex eigenvalues of ${}\varphi$ is smaller than ${}1$.

Proof, Write another proof