Absolute convergence of $ \sum a_nx_n $ implies absolute convergence of $ \sum a_n$

Question

I'm trying to find a proof (or a conunter example, but I'm somehow convinced that the statement is true) for the following fact:

$$ \forall_{(x_n)_{n=1}^{\infty} \lim{x_n} = 0 } \sum\limits_{n=1}^{\infty}|a_nx_n| <\infty \Rightarrow \sum\limits_{n=1}^{\infty}|a_n| < \infty $$

I tried to reach this with contradiction. Suppose $ \sum\limits_{n=1}^{\infty}|a_n| = \infty $ and I aimed to find a sequence $ x_n \rightarrow 0 $ such that $ \sum\limits_{n=1}^{\infty}|a_nx_n| = \infty $, but I don't know how to choose it.

Answer 1

Contradiction is the way to go. Choose $n_1, n_2,...$ such that $$\sum_{n = n_k}^{n_{k+1}} |a_n| > k$$ Then let $x_n = {1 \over k}$ for $n_k \leq n \leq n_{k+1}$.