Let $\displaystyle \sum$$\displaystyle a_{k} from k=1 to infinity be a series of real numbers. Suppose that \displaystyle \sum$$\displaystyle a_{k}$$\displaystyle b_{k} from k=1 to infinity converges for every bounded sequence \displaystyle b_{k}. Prove that \displaystyle \sum$$\displaystyle a_{k}$ from k=1 to infinity converges absolutely.
2. Pick b_k such that $\displaystyle a_k b_k=|a_k|$ for each k. Verify that b_k is bounded (it should be bdd by 1, as you only need to use 1,-1).