Bolzano Weierstrass Theorem (for sequences) states that Every bounded sequence has a limit point. However the converse is not true i.e. there do exist unbounded sequence(s) having only one real limit point. My query is, why are they writing it as "only one real limit point"? I feel it should be "having a (atleast 1) real limit point".
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$\begingroup$ $\{0,1,0,2,0,3...\}$ has only one real limit point, namley $0$. $\endgroup$– geetha290krmCommented Jun 20, 2022 at 4:54
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1$\begingroup$ Could you quote the exact statement that uses "only one real limit point" which you believe is incorrect? $\endgroup$– dxivCommented Jun 20, 2022 at 4:55
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$\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$– Community BotCommented Jun 20, 2022 at 4:56
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$\begingroup$ The Bolzano Weierstrass Theorem states that for any bounded sequence, there exists a convergent $\textbf{subsequence}$. From @geetha290krm example, by taking the odd term only which form a subsequence and constant zero-sequence so it must converging to zero. $\endgroup$– James ChiuCommented Jun 20, 2022 at 5:52
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$\begingroup$ You think the statement should have at least 1 limit point. It is correct for any counter-example of the converse of BWT. But actually, to disprove the converse statement, you only need one simple counter-example like if you find 'an' unbounded sequence which has one limit point only, then you are done ! $\endgroup$– James ChiuCommented Jun 20, 2022 at 5:56
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