text: prove if a^2 and b^2 converges, then ab converge?   i'm assuming that you are talking about two sequences a(sub n) and b(sub n) but are you talking about the sequence converging or the series converging.  the sequence does not necessarily converge. for example, let a = 1, b = -1, the a^2 = b^2 = 1 (for all n) but ab alternates between -1 and +1, so it doesn't converge.  if you are talking about the series where a(sub n)^2 is the nth term, then it goes like this:  if a^2 and b^2 both converge, then you can add the two series together, so (a^2 +b^2) converges, too.  |ab| < max(a^2, b^2) < (a^2 + b^2) so ab converges.