For nonzeros $A,B,C\in M_n(\mathbb{R})$, $ABC=0$. Show $\operatorname{rank}(A)+\operatorname{rank}(B)+\operatorname{rank}(C)\le 2n$ Let $A,B,C\in M_n(\mathbb{R})$ be nonzero matrices such that $ABC=0$.
How can we prove that $\operatorname{rank}(A)+\operatorname{rank}(B)+\operatorname{rank}(C)\le 2 n$ ?
I can prove this for two matrices, but in this case, i can't!

$\newcommand{\rk}{\operatorname{rk}}$
$\newcommand{\im}{\operatorname{im}}$You know that $\im(BC)\subseteq \ker A$. Hence, $n= \rk A+\dim\ker A\ge \rk A+\rk(BC)$.
Now, if you consider the restriction of multiplication by $B$ to the subspace $\im C$ and use rank-nullity theorem, you get $\rk(BC)=\rk C-\dim\ker(B|_{\im C})$. Hence 
\begin{align}
n\ge \rk A+\rk(BC)&=\rk A+\rk C-\dim(\ker B\cap \im C)\ge \rk A+\rk C-\dim\ker B=\\&=\rk A+\rk C+\rk B-n
\end{align}
Whence the thesis.
