Eigenvalues of an idempotent matrix can be $0$ or $1$

Question

Let $A$ be an idempotent matrix and $c$ be an eigenvalue of $A$.
Then there exists a nonzero vector $v$ such that

$Av=cv \Rightarrow A^2v=cAv \Rightarrow Av=c^2v \Rightarrow cv=c^2v \Rightarrow (c-c^2)v=0 \Rightarrow c-c^2=0 \Rightarrow c=0,1$

Is there something wrong with my proof?