$ \sum a_n$ converges absolutely if $\sum a_nb_n $ converges absolutely for all bounded $\{b_n\}$

Question

My question is

Let $\sum a_n $ be a series such that $\sum a_nb_n$ converges absolutely for every sequence $\{b_n\}$ that is bounded. Prove that $\sum a_n$ is absolutely convergent.

My thoughts were as follow.

We know that $\sum a_nb_n$ is absolutely convergent,

$$\sum_{i=1}^n |a_nb_n|$$ is bounded for some $M$.

Since the $b_n$'s are also bounded, then there exists some $K$ such that $K\ge b_n$ for any $n$.

Thus we can write $a_nb_n$ being bounded by $a_n K$, and because of the comparison theorem, since $a_nb_n$ converges absolutely and $a_nK$ converges absolutely, then its must be true that $\sum a_n$ is absolutely convergent.

Is this correct?

Answer 1

Even if you just ask $∑a_nb_n$ to converge (not necessarily uniformly) for every bounded $b_n$, you can take $$b_n = \begin{cases}+1 & a_n \geq 0 \\ -1 & a_n < 0\end{cases} \implies a_nb_n = |a_n|$$

this then says that $\sum |a_n| $ converges, QED.

Answer 2

Responding to question: Taking $b_n=1 ~ \forall n$ makes the result immediate.

Responding to the question of whether your thought is correct: As pointed out by John Ma, you have the "wrong-sided" inequality. For instance, if you took an arbitrary bounded sequence in order to prove the statement, it is bound to fail: the sequence $b_n \equiv 0$ is bounded, makes $\sum a_n b_n$ converge absolutely, but provides no information on absolute convergence of $\sum a_n$.