Let $u \in \mathcal{N}(E)$, with nilindex $p$. Deduce from the previous question that if $p \geq n-1$ and $p \geq 2$ then $\operatorname{Im} u^{p-1} = \operatorname{Im} u \cap \operatorname{Ker} u$ and $\operatorname{Im} u^{p-1}$ has dimension 1.
Let $u \in \mathcal{N}(E)$, with nilindex $p$. Deduce from the previous question that if $p \geq n-1$ and $p \geq 2$ then $\operatorname{Im} u^{p-1} = \operatorname{Im} u \cap \operatorname{Ker} u$ and $\operatorname{Im} u^{p-1}$ has dimension 1.