Nilpotent endomorphism: elementary proof

Question

Let $E$ a finite dimensional linear space with $\dim E=n$ and $u\in\mathcal L(E)$. I need to prove that $u$ is nilpotent iff $u$ is trigonalizable with unique eigenvalue $0$. I know that the proof would be easy if we use JCF and the minimal polynomial or Cayley-Hamilton theorem but the problem is that this question is asked before these theorems have been introduced. Is there an elementary proof?