$\sum_{n=1}^\infty a_nb_n$ convergent for all bounded sequences $b_n$, Prove $\sum_{n=1}^\infty |a_n|$ converge.

Question

I have seen previous questions about this sort of question (assume $\sum_{n=1}^\infty |a_n|$ converge and $b_n$ bounded $\rightarrow \sum_{n=1}^\infty |a_nb_n|$).
But I can't find the answer for this question , perhaps I'm missing something.

I know that $\forall n : |b_n|<M$ for some $M>0$ so $|a_nb_n|\le M|a_n|$
but it doesn't suffice the comparison test since I need to find the inverse inequality symbol $\ge$ and because we don't know if $|a_nb_n|$ converge , only $a_nb_n$ converge for our knowledge.
if we take $b_n = (-1)^n$ then we get that $a_n$ converge by the criteria that $b_n$ is bounded and $a_nb_n$ is convergent.

I'm not sure what else to understand from the meaning of $a_nb_n$ convergent series...

Answer 1

Let$$b_n=\begin{cases}1&\text{ if }a_n\geqslant0\\-1&\text{ otherwise.}\end{cases}$$Then the series $\sum_{n=1}^\infty a_nb_n$ converges. In other words, the series $\sum_{n=1}^\infty|a_n|$ converges.