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Let $a_n$ be a sequence with the property that whenever the sequence $b_n \longrightarrow 0$ ,the series $\sum_\limits{n=1}^{\infty}a_nb_n < \infty $. Prove that $\sum_\limits{n=1}^{\infty}|a_n| < \infty$

Can someone give me a hint for this?

Thank you in advance!

Jose M Serra
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Marios Gretsas
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    For a proof by contradiction, use $b_n=\mathrm{sign}(a_n)/s_n$ where $s_n=\sum\limits_{k=1}^n|a_k|$, as indicated there. – Did Nov 23 '16 at 20:24
  • does bn converges to zero? – Marios Gretsas Nov 23 '16 at 20:27
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    @SimpleArt What? Let me suggest to slow down a notch and to try to actually understand what is written before posting comments such as the preceding one. To be proven: If $(a_n)$ is such that for every $(b_n)$ such that $P$ holds then $Q$ holds, then $R$ holds. Proof suggested in my comment: Let $(a_n)$ such that $R$ does not hold, then there exists $(b_n)$ such that $P$ holds and $Q$ does not hold. Clearer now? – Did Nov 23 '16 at 20:32
  • capo: Same remark as in the comment above. – Did Nov 23 '16 at 20:33
  • Thank you for the explanation @Did – Simply Beautiful Art Nov 23 '16 at 20:48
  • @Did I'm still missing the same thing than capo.You suppose that $R$ does not hold by contradiction. I'm OK here. Then why for $b_n=\mathrm{sign}(a_n)/s_n$, $P$ holds? I imagine that $P \equiv b_n \longrightarrow 0$, right? – mathcounterexamples.net Nov 23 '16 at 21:34
  • @Did why do we derive a contradiction with the sequence bn you indicated above? – Marios Gretsas Nov 23 '16 at 22:33
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    Using Did's hint, the result is immediate upon appealing to the result here – Matematleta Nov 23 '16 at 22:42
  • @mathcounterexamples.net Right. – Did Nov 24 '16 at 05:34
  • @capo Here is a crazy idea: do a tiny little part yourself. At present you have been indicated a full proof, at least tell us how far you went with it (otherwise we might start to have bad ideas about the way you use the site...). – Did Nov 24 '16 at 05:36
  • @Did I should have gone to bed earlier yesterday evening... – mathcounterexamples.net Nov 24 '16 at 07:29
  • @Did check my other questions and then start talking..Also yesterday i did not have the time to think about it.Thank you very much – Marios Gretsas Nov 24 '16 at 14:15
  • "check my other questions and then start talking" Excellent. – Did Nov 24 '16 at 14:33

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