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

My attempt: The partial sums of $\sum a_n$ form a bounded sequence (if not $\sum a_n$ wouldn't converge). Also, $\{b_n\}$ converges to some limit $b$. Suppose $\{b_n\}$ is monotonic decreasing (proof is similar in other case). Then $\forall n$, $b_n - b \geq 0$. Set $c_n = b_n - b$. Then $\{c_n\}$ is monotonic decreasing and $\lim_{n\to\infty}c_n = 0$.

Theorem 3.42 in Baby Rudin: Suppose (a) the partial sums of $\sum a_n$ form a bounded sequence; (b) $b_0 \geq b_1 \geq b_2 \geq \cdots$; (c) $\lim_{n\to\infty}b_n = 0.$ Then $\sum a_n b_n$ converges.

So $\sum a_n c_n$ converges. $\sum a_n c_n = \sum a_n (b_n - b) = \sum a_n b_n - \sum a_n b$ so $$\sum a_n b_n = \sum a_n c_n + b\sum a_n.$$ As the sum of convergent series, $\sum a_n b_n$ converges.

Please point out errors. Thanks in advance :)

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This looks legit to me, I can't find an apparent mistake there.

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