Nilpotent endomorphism/Successive kernels/Fact/Proof

Proof

Let ${}v\in V$. Then, the containment ${}v\in V_{i+1}=\operatorname {kern} \varphi ^{i+1}$ is equivalent with ${}\varphi (v)\in V_{i}=\operatorname {kern} \varphi ^{i}$. This gives the first claim. For the second claim, assume that

${}V_{i}=V_{i+1}\,$

holds for some ${}i<s$. By applying ${}\varphi ^{-1}$, we get

${}V_{i+1}=\varphi ^{-1}(V_{i})=\varphi ^{-1}(V_{i+1})=V_{i+2}\,.$

In this way, we obtain

${}V_{i}=V_{i+1}=V_{i+2}=\ldots =V_{s}=V\,,$

contradicting the minimality of ${}s$.

To fact