Consider a discrete-time system $\sum_{L}^{}$ of the form
$x(k+1) = Ax(k) + Bu(k)$
$y(k) = Cx(k)$
Show that if all the eigenvalues of A are on the open unit disc, show that
$\sum_{L}^{}$ is BIBO stable, i.e. $||y(k)|| ≤ M ∀ k$ if $||u(k)|| ≤ N ∀k$
and
$\sum_{k=0}^{\infty}||u(k)||^2 < +\infty ⇒\sum_{k=0}^{\infty}||y(k)||^2 < +\infty$
Those are the two parts to a proof I'm working on. I understand how to prove that the system is BIBO stable, but I am not sure how to go about proving the second statement in discrete time.