Prove that $S+T$ is nilpotent

Question

Let $V$ be a $K$-vector space and let $S$ and $T$ be two linear operators of $V$. If $S$ and $T$ are both nilpotent and $ST=TS$ then $S+T$ and $ST$ are nilpotent operators

I got this: By hypothesis, $S$ and $T$ are nilpotent operators, which implies that $T^k=0$ and $S^q=0$ for some $k,q \in N$ Then $S^q+T^k=0+0=0$

Therefore $S+T$ is nilpotent, but I am not so sure if my argument is enough to prove what it is required.

Answer 1

Hint: Consider $(S+T)^{q+k}$. Make sure to use that $ST=TS$.