Problem
Suppose $V$ is a vector space and $S,T \in \mathcal{L}(V,V)$ are such that $$\textrm{range} \thinspace S \subset \textrm{null} \thinspace T$$ Prove that $(ST)^2 = 0$
Solution
As $Tv \in V$ we must have $S(Tv) \in \textrm{range} \thinspace S \subset \textrm{null} \thinspace T$ implying $T(S(Tv)) = 0$ for all $v \in V$. Therefore $(ST)^2v = STSTv = S(T(S(Tv))) = S(0) \stackrel{T3.11}{=} 0$ for all $v \in V$, implying $STST = 0$.