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My attempt:

$\lim_{n\to\infty}|a_n||x_n|=0$ , $\lim_{n\to\infty}|x_n|=0$ and so $\lim_{n\to\infty} |a_n|=0$. So $\lim_{n\to\infty} a_n = 0$. Hence the sequence {${a_n}$} is bounded.

Xiangxiang Xu
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Antimony
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    Your attempt makes no sense. If $a_n=n$ and $x_n=1/n^2$ then the first two results are true but $a_n\not\to0$. – Peter Foreman May 03 '20 at 08:33
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    The constant sequence $a_n=1$ is a sequence such that $\sum_{k=1}^\infty \lvert a_kx_k\rvert<\infty$ for all sequences ${x_k}{k\in\Bbb N}$ such that $\sum{k=1}^\infty\lvert x_k\rvert<\infty$. –  May 03 '20 at 08:33
  • https://math.stackexchange.com/questions/2040961/does-sum-limits-i-1-inftya-ix-i-infty-whenever-sum-limits-i-1?rq=1 –  May 03 '20 at 11:30

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