Let $T \in \text{Hom}_F(V,V)$, suppose the characteristic polynomial of $T$, $c_T(x) = (x- \lambda)^kp(x)$, where $p(\lambda) \neq 0$, show that $\text{dim}_F (E_{\lambda}^\infty) = k$, where $E_{\lambda}^\infty = \bigcup_{i = 1}^{\infty} \text{ker}(T-\lambda I)^i$.
This is a common result however, I can't seem to find simple clear proof.