Since $y$ is eigenvalue of $A$ it means there is $\bold{0}\not =\bold{x}\in \mathbb{R}^n$ so that $A\bold{x} = y\bold{x}$.
Thus, $A(A\bold{x}) = A(y\bold{x}) \implies A^2 \bold{x} = y(A\bold{x})\implies \bold{0} = y(y\bold{x}) \implies y^2\bold{x} = 0$
From here we see that $y=0$ because $\bold{x}\not = \bold{0}$.