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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