Showing that $O$ is the only nilpotent matrix in $\langle A \rangle$ where $A$ is diagonalizable I have the following task: 

Let $A\in \mathcal{M}_n(K)$ be a diagonalizable square matrix. Show
  using the spectral decomposition of $A$  that the only nilpotent
  matrix in $\langle A\rangle =\{p(A)=c_0A^k+\cdots+c_{k-1}A+c_kI\mid
 c_i\in K\}$ is zero matrix $O$. $K$ is a field.

The spectral decomposition of $A$ is $A=\sum_{i=1}^n \lambda_i E_i,$
where $\lambda_i$s are the eigenvalues of $A$ and the matrices $E_i$ are idempotent pairwise orthogonal matrices. 
I have tried to solving this task, but I'm in a dead end at the moment. Thank you for your help!

Hint: show that all 


*

*matrices in $\langle A\rangle$ are diagonalizable, and that

*the only nilpotent diagonalizable matrix is the zero matrix.
