Suppose $V$ is a vector space and $S,T \in \mathcal L(V,V)$ such that

Prove that $(ST)^2 = 0$.

Let $v \in V$, consider how $S(Tv) \in \text{null }T$ so $T(STv) = 0$, and thus $(ST)^2v = 0$ for any $v \in V$.