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?
Best Answer
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.