Column and Null space proof!

The question is :

Let

A be an m×n matrix and B an n×r matrix. If AB = O (the m×r matrix of zeroes)

prove that the column space of B is contained in the null space of A.

Here is what I have gotten:

We know:

Where the null space is the span of the set of all vectors x that satisy the equation. Let null space(A) = span( ) I let col. space(B)=span( )

Let v be in the col. space(B)

Therefore where are elements of

If we can show that v can be shown to be apart of the span( ) then we are done. I cant think of a way to go about doing this.

Any help would be greatly appreciated!