# If $\{a_nb_n\}$ is absolutely convergent for every $\{b_n\}$ that converges to 0, then $\{a_n\}$ is absolutely convergent. [duplicate]

Let $\{a_n\}$ be a sequence such that $\{a_nb_n\}$ is absolutely convergent for every $\{b_n\}$ that converges to 0. Prove that $\{a_n\}$ is absolutely convergent.

My thought was that since the $b_n$ converge to 0, then their terms are also getting small.

Thus, for large n, the $a_nb_n$ terms are small such that they absolutely converge.

Is this valid?

We are in Reals 2, have yet to learn about Banach spaces.

## marked as duplicate by Adam Hughes, Empty, Mankind, user99914, PhoemueXOct 1 '15 at 12:22

• Hello and welcome! Your reasoning has a flaw. You must show that the $a_n$ converge - the convergence of the sequence of $a_nb_n$ is assumed. And what do you mean when you say the sequence is " absolutely convergent"? Do you perhaps mean the series? – Hans Engler Oct 1 '15 at 3:31
• I assumed he is talking about the series $\sum a_n b_n$. This question is a duplicate, I will flag it soon. – Olorun Oct 1 '15 at 3:32