If a sequence $a_n$, $n\in\mathbb{N}$ doesn't have any convergent subsequence can $|a_n|\rightarrow a$, $a\in[0,\infty)$?

My intuition says that this isn't possible but I'm not sure how to prove it..

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    $\begingroup$ Well, if the absolute sequence converges then so does the sequence of positive terms and the sequence of negative terms (one of which, at least, must be infinite). $\endgroup$ – lulu Jan 21 at 20:32
  • $\begingroup$ Hint: do you know any conditions which imply that $a_n$ does have a convergent subsequence? Your sequence must violate those conditions. $\endgroup$ – Wojowu Jan 21 at 20:32

If $(|a_n|) $ is convergent, then $(a_n) $ is bounded. Now invoke Bolzano-Weierstraß. Conclusion?

  • $\begingroup$ Is there any specific theorem that suggests so? $\endgroup$ – SlimJim Jan 22 at 19:07

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