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I know $\mathbb{R}$ is complete since every Cauchy sequence of numbers has a limit. But does this limit need to be in the same metric space as the sequence. For example is $\mathbb{Q}$ complete? Every Cauchy sequence has a limit, but not necessarily in $\mathbb{Q}$, just take $x_n = (1+\frac{1}{n})^n$.

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    $\begingroup$ Completeness supposes the limit is in the same space. Any metric space (actually any uniform space) has a completion, so, if the limit were allowed to be ioutside the metric space, any space would be complete. $\endgroup$ – Bernard Nov 4 '15 at 14:32
  • $\begingroup$ So then $\mathbb{Q}$ is not complete? $\endgroup$ – continental Nov 4 '15 at 14:33
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    $\begingroup$ @dable, that is correct. $\endgroup$ – Paul Nov 4 '15 at 14:38
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No, $\mathbb{Q}$ is not complete because it does not contain all its limit points.

It is easy to transform any metric space into a complete metric space by adding in the limit points. More precisely, one can consider the set of Cauchy sequences on a metric space under the equivalence relation that two Cauchy sequences are the same if their difference converges to $0$. This is one way of completing a space. For instance, the real numbers can be constructed from $\mathbb{Q}$ in this way.

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Yes. Generally, something is complete if it doesn't need anything added to it. In this particular sense of complete, $\Bbb R$ is complete because limit points for Cauchy sequences in $\Bbb R$are already in $\Bbb R$ and so do not need to be added, while $\Bbb Q$ is not complete because limit points for Cauchy sequences in $\Bbb Q$ do not in general belong to $\Bbb Q$.

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  • $\begingroup$ Careful with the phrasing in these answers. The precise meaning of "limit point" is relative to the space you are in. So the space $Q$ does, in fact, contain all of its limit points as does any metric space. The only way I know to say this correctly is to say that "there are, however, Cauchy sequences in $Q$ that do not converge in $Q$." $\endgroup$ – B. S. Thomson Nov 4 '15 at 17:01
  • $\begingroup$ @B.S.Thomson: You are right. I realize that the wording is still not ideal from a constructive perspective, but I am trying to link the word complete to the everyday meaning, where what is needed to complete something is lying around already and doesn't have to be created out of of equivalence classes of sequences of elements of the original thing. $\endgroup$ – John Bentin Nov 4 '15 at 19:34

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