what you want to show is that:
rank(A) + rank(B) ≤ rank(A+B).
but this is not true. suppose that A =
[1 0]
[0 0] and that B =
[-1 0]
[ 0 0].
clearly, A and B both have rank 1, so rank(A) + rank(B) = 1 + 1 = 2.
however, A+B is the 0-matrix, which has rank 0, and 2 is not less than or equal to 0.