prove $Null(A)\subset Null(A^2)$
2. Well if $x \in \mbox{Nullspace }(A)$, then $Ax=0$, so also $A^2x = A(Ax)=A(0)=0$, so $x \in \mbox{Nullspace }(A^2)$. Therefore $\mbox{Nullspace }(A) \subset \mbox{Nullspace }(A^2)$.