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Let $(X,d)$ be a metric space and $(x_n)$ be a Cauchy sequence in $X$. Is there a limit $x$ for $(x_n)$ whether $x$ in X or not ?

In general, does every Cauchy sequence has a limit, if that limit point lies in the same metric space which Cauchy sequence defined, then that Cauchy sequence converges otherwise it's divergent?

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  • $\begingroup$ I'd say that you should check carefully your question's redaction as it seems to me that the way it is posed now it is pretty messy. $\endgroup$ – Timbuc Apr 9 '15 at 14:01
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    $\begingroup$ "Whether in $X$ or not" ... perhaps that is the crux of the question. Certainly the sequence converges to something in some space $Y \supset X$, such as the completion of $X$. $\endgroup$ – GEdgar Apr 9 '15 at 15:07
  • $\begingroup$ @GEdgar I think that this is what OP asks for actually. $\endgroup$ – GPerez Apr 9 '15 at 22:08
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A Cauchy sequence in a metric space $(X,d)$ is not necessarly convergent. for example, consider the metric space $(\mathbf{Q},d)$ where d is the usual absolute value. And consider the sequence $x_{n}$, $x_{1}=1$, $x_{n+1}=\frac{1}{2}(x_{n}+\frac{2}{x_{n}})$. One can check that $x_{n}$ is a Cauchy sequence in $\mathbf{Q}$ but has no limit in $\mathbf{Q}$.

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  • $\begingroup$ Epic example! +1. $\endgroup$ – Hasan Saad Apr 9 '15 at 14:45
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I'm not really sure what your question even means. The way it's written is dubious.

However, there is a short answer to this. No. Not every cauchy sequence converges. They only do if they are in a complete metric space.

Take, for example, the sequence $\frac{1}{n}$ in the metric space (open interval) $(0,1)$. It is obviously Cauchy, however, it has no limit in such a space.

For the sake of completeness, you should read on this subject on wikipedia if it is new to you:

http://en.wikipedia.org/wiki/Complete_metric_space

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