Suppose that the summation from k=1 to infinity of a_k converges absolutely. Prove that the series summation from k=1 to infinity of (a_k)^2 converges.
Problem: Suppose that converges absolutely. Prove that converges as well.
Proof: Since, by definition there exists some such that . Therefore, for . Therefore , and since converges absolutely, does as well. Therefore, since both of which converge it is clear that converges.