TITLE: Does $\sum |a_n|b_n $ converge?
QUESTION [5 upvotes]: Let $$\sum |a_n|<\infty,\quad \sum b_n<\infty$$
Then
$$\sum |a_n|b_n < \infty?$$

I tried to bound this sum, since $0 < a_n < 1 $ for $\forall n>N$
but this wasn't efficient because $b_n$ isn't a positive sequence.
Next, I tried to use AM-GM but I don't have $\sum (b_n)^2<\infty$ so I guess I need another approach to prove / disprove this one.
Thanks for help

REPLY [5 votes]: We simply have that eventually $|b_n|<1$ and then $\sum |a_n|b_n$ converges absolutely by direct comparison test with $\sum |a_n|$.