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.