Proving Eigenvalue squared is Eigenvalue of $A^2$

We know $Ax = \lambda x$. Then $A \lambda x = \lambda(Ax) = \lambda^2x = A^2x$. Putting this into a more readable mathematical sentence, we get:

$$A^2x = A(Ax) = A\lambda x = \lambda(Ax) = \lambda^2x$$

You were done and didn't realize it. :)

You are on the right way: let $x$ an eigenvector of $A$ associated to the eigenvalue $\lambda$ so $$Ax=\lambda x$$ and then apply $A$ we find $$A(Ax)=A^2 x=A(\lambda x)=\lambda A x=\lambda\lambda x=\lambda^2 x$$ and conclude.