Suppose that $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Prove that the null space of $B$ is a subset of the null space of $AB$, that is $\nsp{B}\subseteq\nsp{AB}$. Provide an example where the opposite is false, in other words give an example where $\nsp{AB}\not\subseteq\nsp{B}$.