Let $\displaystyle A \in M_{n}(R)$ such that $\displaystyle A^{m}=0$ for some positive integer $\displaystyle m$. If $\displaystyle A$ is a nonzero matrix, show that $\displaystyle A$ is not diagonalizable.

I proved that if $\displaystyle A^{m}=0$ for some positive integer $\displaystyle m$, then the only eigenvalue of $\displaystyle A$ is $\displaystyle 0$. I think it can be proven by contradiction. But I'm still struggle.