I know about the fact that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$, where $A$ and $B$ are $m \times n$ matrices.
But somehow, I don't find this as intuitive as the multiplication version of this fact. The rank of $A$ plus the rank of $B$ could have well more than the columns of $(A+B)$! How can I show to prove that this really is true?