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