Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 3, Problem 8

Problem

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

Answer

Since $b_n$ is monotonic and bounded, $b_n \rightarrow b$. If $b_n$ is increasing, then $\sum a_n (b-b_n)$ converges by Theorem 3.42. Then $\sum a_nb_n = \sum a_n b - \sum a_n (b - b_n)$ also converges.

If $b_n$ is decreasing, then $\sum a_n (b_n-b)$ converges by Theorem 3.42. Then $\sum a_nb_n = \sum a_n b + \sum a_n (b_n - b)$ also converges. 

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