Analysis: Given a series $a_n$ and a sequence $c_n$ prove $\sum_{n=1}^{\infty}a_nc_n$ converges.

Question

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.

Answer 1

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.

Answer 2

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 $\sum c_na_n < \infty.$