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Prove that if $\sum |a_n|$ converges, then $\sum a_n^2$ also converges.

Attempt:

$\sum |a_n|$ converges $\implies \sum |a_n|<M$.

If $\sum |a_n|$ converges, then $\sum a_n $ also converges.

Could anyone please tell me on how to proceed ahead with regard to this problem.

Thank you very much for your help

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marked as duplicate by Git Gud, copper.hat, leo, Lord_Farin, user98602 Jan 26 '15 at 18:18

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    $\begingroup$ $\sum a_n^2 \le (\sum |a_n|)^2$,bounded monotone convergence thm $\endgroup$ – r9m Jan 26 '15 at 17:48
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    $\begingroup$ After a while $|a_n|\lt 1$, so $a_n^2\lt |a_n|$. $\endgroup$ – André Nicolas Jan 26 '15 at 17:49
  • $\begingroup$ Thank you for your answers. I just lost it all in this problem, I guess. $\endgroup$ – MathMan Jan 26 '15 at 17:52
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    $\begingroup$ Note that $|a_n|$ must be bounded by some $M$ and so $|a_n|^2 \le M |a_n|$. $\endgroup$ – copper.hat Jan 26 '15 at 17:52
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Prove that there is a tail of the summation for which $a_n^2<|a_n|$ holds for all terms in the tail.

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