Show there exist p such as $Kerf \varsubsetneqq Kerf^2\varsubsetneqq ...\varsubsetneqq Kerf^p=Kerf^{p+1}$ Let E a finite dimensional vector space ($dimE = n \in \mathbb{N^*}$) and $f\in \mathcal{L}(E)$ How to show that there exist $$p\in [\\![0,n]\\!] \ \text{ such that:} \ \\{0\\}\varsubsetneqq Kerf \varsubsetneqq Kerf^2\varsubsetneqq ...\varsubsetneqq Kerf^p=Kerf^{p+1} \ ?$$

The reason is that otherwise you would have an **infinite nested sequence** of subspaces, each one with at least one dimension less than the previous one, which is impossible because of the finite dimension of space $E$.

(I assume that, besides, you know how to prove that for any $k$, $Kerf^k \subset Kerf^{k+1}$ ).