# real analysis – Proving \$sum_{k=1}^infty a_kb_k\$ converges given certain properties

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!