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Problem: If $\sum a_n$ converges, and $\{b_n\}$ is monotonic and bounded, prove that $\sum a_n b_n$ converges.

Source: Rudin, Principles of Mathematical Analysis, Chapter 3, Exercise 8.

2 Answers 2

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The sequence $\{b_n\}$ is monotonic and bounded, so it converges to some number $C$. Assume, without loss of generality, that the sequence $\{b_n\}$ is increasing, and write $b_n=C-d_n$, where $d_n\rightarrow 0$. We have

$\sum a_nb_n = C\sum a_n -\sum a_nd_n.$

The first series on the right is convergent by hypothesis, and the second is convergent because of the following theorem:

Theorem: If the partial sums of $\sum t_n$ form a bounded sequence and $s_n$ is a decreasing sequence that tends to 0, then $\sum t_ns_n$ converges.

Here we take $t_n=a_n$ and $s_n=d_n$.

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    @DanDouglas It's slightly tricky. You need to use summation by parts. The following 2 images are excerpts from Rudin's book, pages 70 and 71. [Click here.](http://imgur.com/a/ixKBy) Let me know if they answer your question or if you need more detail.2013-08-24