Theorem:
Let $L$ be a subalgebra of $\mathfrak{gl}(V)$ with $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V\ne 0$, then $\exists v\in V. v\ne 0$, such that $Lv = 0$.
Humphreys gave the following proof on page 13:
I failed to see how the underlined statement, albeit correct, relates to the proof. Why did Humphreys put this sentence here? Am I missing something?
If $W$ is not fixed by the action of $z$ you can not find the eigenvector for $z$ in $W$ (because it can properly act on $V$ but not on $W$)