|| A^K ||_2 = max_{x !=0} [||A^K x||_2 / ||x||_2]

................ = max_{x!=0}|| [A ( A^{K-1} x||_2 / ||X||_2]

................ < k max_{x!=0}|| [ A^{K-1} x||_2 / ||X||_2]

for some k such that: ||A||_2 < k < 1, hence:

|| A^K ||_2 < k^K

But as K -> infty, k^K ->0, so

Lim as k-> infinity, ||A^K||_2 = 0

hence A^K -> 0 (as in this case convergence in norm implies convergence elementwise

though I'm not sure if you will have to prove this).

RonL