I'm not sure how to do this. . . .
Prove that, for any m x n matrix A and any n x p matrix B, the column space of AB is contained in the column space of A.
Here's a hint:
If we think of any (m x n) matrix M as a linear transformation from R^n to R^m, then the column space of M is precisely the image of M... that is, if I take a column vector v in R^n, then the vector Mv in R^m always lies in the column space of M.