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My book states that

if a sequence $a_n$ converges to $l$, then every subsequence also converges to $l$.

Now it states at the end that its converse is not true. but as we know that if every subsequence converges to same limit $l$ then $a_n$ also converges to $l$. Kindly tell me if am wrong in understanding the converse or the author has made any mistake. There should be iff condition in the above theorem.

paulina
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Fayaz' Rasool
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    Pretty sure they mean that if a specific subsequence tends to $l$, you can't conclude that $a_n$ tends to $l$ as well. (nor that it converges at all) – Mark May 25 '24 at 09:39
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    @Mark pointed out one possible correct interpretation. Note that another similar result which is implied by this but whose converse is not true is "If $a_n\to l$ then every convergent subsequence of $a_n\to l$". The converse of this is not true since a divergent sequence may have no convergent subsequence. – Carlyle May 25 '24 at 12:20
  • You might also be interested in the "subsequence principle" explained here: https://math.stackexchange.com/questions/397978/every-subsequence-of-x-n-has-a-further-subsequence-which-converges-to-x-the – PhoemueX May 25 '24 at 15:17

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