How to prove that if A and B are matrices, then the rank of AB <= the rank of B So far I know that when matrices A and B are multiplied, with B on the right, the result, AB, is a linear combination of the columns of A, but I'm not sure what to do with this. 

The image of $B$ is a subspace of dimension $\mathrm{rank}(B)$. Left multiplication by $A$ transforms this into a new subspace, which is the image of $AB$ having dimension $\mathrm{rank}(AB)$; this linear transformation cannot increase the dimension of the subspace.
Put in other words, $\mathrm{rank}(B)$ is the dimension of the space generated by the rows of $B$, and $\mathrm{rank}(AB)$ is the dimension of the space generated by the rows of $AB$. The rows of $AB$ are linear combinations of the rows of $B$, so the dimension of this space is restricted by $\mathrm{rank}(B)$ (the columns of $AB$ are also linear combinations of the columns of $A$ as you mention, but this fact is less helpful here).
