# Convergence of $\sum a_n b_n$

In Rudin P.M.A

The partial sums $A_n$ of $\sum a_n$ form a bounded sequence; i.e.

$b_0\ge b_1\ge b_2\ge\cdots\ge b_n$ so that $\lim\limits_{n\rightarrow\infty}b_n=0$.

Then $\sum a_n b_n$ converges.

To prove this he used partial summation formula.

My question is that if we are given that $\sum a_n$ converges, and if {$b_n$} is monotonic and bounded, then can I say $\sum a_n b_n$ converges, since

$$\sum a_n b_n\le\sup{b_n}\sum a_n$$

or do I have to use Rudin's theorem?

• The inequality you plan to use holds provided every $a_n$ is nonnegative -- a hypothesis which is not made. – Did Jun 12 '16 at 18:40
• Ohhhhh.. Thanks! – Xaviere Jun 12 '16 at 18:57
• See Dirichlet's Test in Wikipedia. The def'n there has the $a_n$ and $b_n$ interchanged with respect to your notation. – DanielWainfleet Jun 12 '16 at 23:47