We have two sequences ${a_n}$ and ${b_n}$. We know that $a_n to 0$, $sum_{n=1}^infty |a_{n+1}-a_n| < infty$, and that $exists M>0$ such that $|sum_{n=1}^N b_n| leq M ; forall N in mathbb{N}$. How can we show that $sum_{n=1}^infty a_nb_n$ converges?

My idea is the following: Let $S_k=sum_{n=1}^k a_nb_n$ and $B_k=sum_{n=1}^k b_n$. We want to show that $S_k$ converges. From summation by parts, we have $S_k=a_kb_k+sum_{n=1}^{k-1} B_k(a_k-a_{k-1})$. The first term goes to $0$ as $k to infty$, so we just need to show $sum_{n=1}^{k-1} B_k(a_k-a_{k-1})$ also converges.

This is where I’m stuck, but I suspect it’s possible to show this. Thanks!