http://img28.imageshack.us/img28/5227/79425145.jpg

This is my attempt at a solution:

A^2(x) = A(A(x))

A^2(x) = lambda^2(x)

Therefore x is an eigenvector of A^2 with eigenvalue lambda^2. The statement for general N>0 follows in the same way.

Am I missing anything?