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Here's the question:

Let $\sum_{n=1}^{\infty}a_n$ be a convergent series with positive terms. Prove that, if $c_n$ is a sequence of positive terms satisfying $\lim_{n \rightarrow \infty}c_n=0$, then $\sum_{n=1}^{\infty}a_nc_n$ converges.

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  • $\begingroup$ Was it an actual question? If so, why did you answer it yourself immediately -- what was the intention behind that? $\endgroup$ – Clement C. Dec 8 '15 at 16:59
  • $\begingroup$ @ClementC. It was an "actual" question for a problem sheet that I was given. I answered it myself immediately as there is an option (when posting a question) to post your own answer "Q&A style". I am using this platform to help revise for my studies and so posting questions to which I have often planned an answer helps fortify the concepts in my brain. Apologies if this bothers you. $\endgroup$ – Geometry Dec 8 '15 at 17:17
  • $\begingroup$ OK -- it does not bother me, mostly puzzled me. $\endgroup$ – Clement C. Dec 8 '15 at 17:56
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Let $S_k=\sum_{n=1}^{k}a_n$ be the $k^{th}$ partial sum of $\sum_{n=1}^{\infty}a_n$. Since $\lim_{n \rightarrow \infty}c_n=0$ and $c_n>0$ for all $n$, there exists an $n_0>0$ such that $0<c_n<1$ for all $n>n_0$.

Let $\space M = max(1, c_1, c_2, \space ...,\space c_{n_0})$, and note that $c_n \leq M, \space \forall \space n \geq 1$. In particular, the $k^{th}$ partial sum of the series $\sum_{n=1}^{\infty}a_nc_n$ satisfies

$$\sum_{n=1}^{k}a_nc_n \leq \sum_{n=1}^{k}a_nM=M \space S_k.$$

Hence, the sequence of partial sums of the series $\sum_{n=1}^{\infty}a_nc_n$ forms a monotonic (since all the $a_n$ and $c_n$ are positive), bounded (above by $M\sum_{n=1}^{\infty}a_n$, and below by $0$) sequence, and so converges. That is, $\sum_{n=1}^{\infty}a_nc_n$ converges.

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I'm puzzled about the assumption $c_n \to 0.$ It's not necessary. Just assume the sequence $c_n$ is nonnegative and bounded, say $c_n\le M$ for all $n.$ Then $0\le c_na_n \le Ma_n$ for all $n$ and $\sum Ma_n < \infty.$ Hence $\sum c_na_n < \infty.$

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