Question

I have a real analysis math question. if a_n is a sequence such that the summation of b_n from n=1 to infinity converges whenever b_n is a sub-sequence of a_n, the the summation of a_n from n=1 to infinity converges absolutely. I know the if every subsequence converges then the whole thing must converge, but I'm not sure how to prove it. I thinking maybe using contradiction. Riemann rearrangement?