Verify that the image of $\psi$ is contained in the kernel of $\xi^n$.

Prove that the kernel of $\psi$ is a complement of $W$ stable by $u$.

Let $u$ be a nilpotent endomorphism of a finite-dimensional vector space $V$. Prove that there exists a basis of $V$, a natural integer $s$ and nonzero natural integers $r_1 \geqslant \cdots \geqslant r_s$ in which the matrix of $u$ is block diagonal and whose diagonal blocks are Jordan blocks $J_{r_1}, \ldots, J_{r_s}$ of respective sizes $r_1, \ldots, r_s$.