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. 

Comments

Popular posts from this blog

Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 2, Problem 14

Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 5, Problem 15

Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 4, Problem 4