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Rudin 3.2(d): If $X$ is a metric space and $E$ is a subset of $X$ and $p$ is a limit point of $E$, then there is a sequence $\{p_n\}$ in $E$ such that $\{p_n\}$ converges to $p$.

Why the converse: if there is a sequence converging to $p$, then $p$ is a limit point, is not generally true?

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  • $\begingroup$ It's definitional. Some authors will say that every element of $E$ is a limit point of $E$, precisely because of the constant sequence. To answer your question precisely, we need to know which definition of limit point you're using. $\endgroup$ – dbx Nov 20 '17 at 17:23
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Let p be a non limit point. Thus the sequence $\{x_n\}_{n\in\mathbb N},\ x_n=p\forall n\in \mathbb N$ converges to p.

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  • $\begingroup$ Thanks. If I add the restriction that every term of the sequence does not equal p, can I draw the conclusion that p is a limit point? $\endgroup$ – Zhou Chenliang Nov 20 '17 at 17:13
  • $\begingroup$ @ZhouChenliang No, you'd need "an infinite number of elements of the sequence are no equal to $p$." Or, equivalently, the sequence is not ultimately equal to $p$ identically. $\endgroup$ – Clement C. Nov 20 '17 at 17:27
  • $\begingroup$ @ClementC. I think my restriction is stronger then “an infinite number of elements of the sequence are not equal to p”? $\endgroup$ – Zhou Chenliang Nov 20 '17 at 17:31
  • $\begingroup$ @ZhouChenliang The quantifiers/phrasing are confusing. Did you mean "not(every term of the sequence is equal to $p$)" (not every term of the sequence is equal to $p$, as I read it), or "every term of the sequence is not equal to $p$" ("no term of the sequence is equal to $p$")? $\endgroup$ – Clement C. Nov 20 '17 at 17:34
  • $\begingroup$ @ClementC. Well sorry about that. I meant no term is equal to p $\endgroup$ – Zhou Chenliang Nov 20 '17 at 17:52
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If the sequence $(p_{n})$, $p_{n}\ne p$, given $\epsilon>0$, choose some $N$ such that $p_{n}\in B_{p}(\epsilon)$ for all $n\geq N$. So $p_{n}\in B_{p}(\epsilon)-\{p\}$ for all such $n$, the neighbourhood even contains infinitely many $p_{n}$, so it is a limit point.

As long as the sequence does not eventually equal to $p$, then it goes through.

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