If $\varphi$ is an Endomorphism over $V$ with $\mu_\varphi=\prod^n_{i=1}p_i^{m_i}$ for some $n,m_i$ and irreducible $p_i$ and $\chi_\varphi=\prod^n_{i=1} p_i^{c_i}$ with $c_i\geq m_i$ then $\operatorname{ker} p_i^{m_i}(\varphi)=\operatorname{ker} p_i^{c_i}(\varphi)$. Why? This statement is required to prove, that generalized eigenspaces can be optained from either the minimal polynomial or characteristical polynomial.
I get why $\operatorname{ker} p_i^{m_i}(\varphi)\subseteq\operatorname{ker} p_i^{c_i}(\varphi)$ (this directly follows from $m_i\leq c_i$). I don't get why the other direction holds though.