Let $A$ be a $n$ x $n$ matrix with complex entries. Define $m=|${$\text{rank}(A^k)$ $-$ $\text{rank}(A^{k+1})$| k$\in$ $\text{N*}$}$|$. Prove that $n+1$ $\ge$ $\frac{m(m+1)}{2}$.
I have tried writing $A$ in its Jordan form and look at the block matrices that have $0$ in them, but it looks like there are too many values of the dimensions of the blocks to keep count of when raising the matrix to different powers. I would love to know a solution using this idea.