Prove $rgA + rgB \le n$

Question

Let $A,B\in \mathbb{R}^{n\times n}$. $\exists a_{i,j},b_{k,l} \not = 0$.$$A \times B = O$$ ($O$-zero matrix). Prove $$rgA + rgB \le n$$ My attempt: as rows of matrix $AB$ are linear combinations of rows $B$, so we have $$a_{i1}B_{1}+ \ldots +a_{in}B_{n}= \theta, i=\overline{1,n},$$ $B_1, \ldots ,B_n$ - rows of matrix $B$. As $A$ is non-zero matrix there is $B_s$, which is linear combination of other rows of $B$, hence system $B_1, \ldots ,B_n$ is linearly dependent, so $rgB < n$. And I got stuck. Is there a way to imrove that estimate?