Let $L: V\to V$ be an operator in a finite-dimensional vector space $V$ over $R$. For any $n \geq 0$, let $K_n = \ker (L^n)$, $I_n = \mathrm{Im}(L^n)$.
(a) Prove that there exists $N$ such that for all $n \geq N$, we have $K_n=K_N$ , $I_n= I_N$.
(b) Denote $K = K_N$, $I = I_N$, where $N$ is the same as above. Prove that $LK$ is contained in $K$, and $LI$ is contained in $I$, and the restriction of $L$ to $K$ is nilpotent, restriction of $L$ to $I$ is invertible.
(c) Prove that $V = K \oplus I$.
(For part c, We assume without proof that if $p \in R[x]$ is the characteristic polynomial of $L$, then $p(L) = 0$ , how to proceed?)