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I want to know the example of a sequence $x_n$ such that there is no convergent subsequence but $|x_n|$ converges.

My attempt:I think there will be no such sequence as if $\lim(|x_n|)$ is not $\infty$ then that would mean that $|x_k|<M $ for $k \ge N$ .Hence the sequence $x_n$ will be bounded so by bolzano weistrass it will have a convergent subsequence.

Antimony
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    This seems correct to me. There was a similar question here: https://math.stackexchange.com/questions/436120/real-analysis-a-sequence-that-has-no-convergent-subsequence – Phil May 03 '20 at 05:25
  • Hint: Consider the subsequence of only non-negative terms (or the subsequence of only non-positive terms). Those terms are all $x_{i_k} = |x_{i_k}|$ so this is a subsequence of a converging sequence. – fleablood May 03 '20 at 05:28
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    Your approach is correct. – Naba Kumar Bhattacharya May 03 '20 at 05:28

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