If for linear operator $A:E\to E$, $A^k=0$, then $A^n=0$

Question

Let for the linear operator $A:E\to E$, $A^k=0$, then $A^n=0$.

Here $n=dim E$ and $n <k.$

I tried to find the characteristic oolynomial that I think it is the clue, however did nit succeed.

Any suggestion?