Proof of Jordan normal form

Question

I want to prove Jordan Decomposition Theorem, but I encounter a problem: Let $A$ be a $n \times n$ nilpotent matrix with $A^m=0$, there exists an invariant subspace $W=\text{Span}(v,Av,...,A^{m-1}v)$ of V, let $V=W\bigoplus W’$, so if $W’$ is an invariant subspace of A?

Answer 1

Not necessarily. Let $A$ act on a basis like $e_1\mapsto e_2,\ e_2,e_3\mapsto 0$, and let $v:=e_1$, thus $W=\mathrm{span}(e_1,e_2)$.
Now, with $u:=e_1+e_3$, set $W':=\mathrm{span}(u)$. Then we have $V=W\oplus W'$, but $Au=e_2\notin W'$.