if $(a_{n})$ is bounded and $\sum(b_{n})$ is absolutely convergent, then $\sum(a_{n}b_{n})$ is absolutely convergent

Question

How do I prove that if $(a_{n})$ is bounded and $\sum(b_{n})$ is absolutely convergent, then $\sum(a_{n}b_{n})$ is absolutely convergent?

I tried to consider the sequence of partial sums of $\sum |a_nb_n|$

$S_n=|a_1b_1|+|a_2b_2|+......+|a_nb_n|$

And so $S_n \le M (|b_1| + ... + |b_n|)$ for a fixed $M$, so.... — , Sep 12 '17 at 22:34
If you understand this answer, you should be able to answer your question. — Git Gud, Sep 12 '17 at 22:37

score 0 · Accepted Answer · answered Sep 12 '17 at 22:47

0

hint

$(a_n) $ bounded $$\implies (\exists M\ge 0)\;\;(\forall n\in \Bbb N)\;\; |a_n|\le M $$

$$\implies |a_nb_n|\le M|b_n|$$

and $\sum |b_n|$ convergent $\implies \sum M|b_n|$ convergent.

You conclude by comparison test.

answered Sep 12 '17 at 22:47

hamam_Abdallah

1 Answers1