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?
A proof by induction works like this: first show your statement holds for or or the first for which it is true, then assume it to be true for (this is your induction hypothesis) and then show it to be true for using your hypothesis. if is an eigenvalue of , and so is an eigenvalue of for . suppose it is true for ,then
it is essentially what you said but a a bit more rigorous
induction is a really useful tool, you should practice using it