Nilpotent endomorphism/Successive kernels/Fact/Proof

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

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

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

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

In this way, we obtain

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

contradicting the minimality of ${\displaystyle {}s}$.