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The standard definition of a complete metric space is that all Cauchy sequences converge in the space. It's easy to come up with examples where a Cauchy sequence has a natural limit that simply is not in the space (e.g. any sequence in $\mathbb{Q}$ that should converge to an irrational number). However, this way of thinking about it can be confused with closure. Typically, I try to differentiate the two by thinking about completeness as more a property of the metric than a property of the space: in a complete metric space, any sequence that doesn't have a limit won't be Cauchy because the metric will be more "appropriate" in some sense. To further differentiate between completeness and closure, I'm wondering if there are examples of Cauchy sequences in incomplete metric spaces that don't have a "natural limit" in a larger space, though this seems unlikely to me. (I ask this because it's a different way that a Cauchy sequence wouldn't converge in the space -- rather than converging to something outside the space, it wouldn't converge at all.)

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    $\begingroup$ You should search about the completion of a metric space $\endgroup$ – Evangelopoulos F. Apr 10 at 15:52
  • $\begingroup$ I'm familiar with the completion of a metric space. I like to think of the completion of a metric space as a space of equivalence classes of sequences where sequences are equivalent if they converge to the same value, and elements in the original space can be thought of as constant sequences. I suppose another way to think about the completion is just by adding limits of Cauchy sequences, though that sort of begs the question. $\endgroup$ – rem Apr 10 at 15:58
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    $\begingroup$ Actually, that does not beg the question, that answers the question, because you can define equivalence classes of Cauchy sequences without every invoking the concept of "convergence to the same value". That's kind of the whole point of the construction of the completion of a metric space. $\endgroup$ – Lee Mosher Apr 10 at 16:03
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    $\begingroup$ Also, your example of $\mathbb Q$ is not quite apt. One of the common axiomatic constructions of $\mathbb R$ is, simply, as the completion of $\mathbb Q$ (i.e. one defines Cauchy sequences in $\mathbb Q$, and equivalence classes of Cauchy sequences, purely in terms of the $\mathbb Q$-valued metric $d(r,s)=|r-s|$). $\endgroup$ – Lee Mosher Apr 10 at 16:05
  • $\begingroup$ @LeeMosher, yes I understand that the equivalence class space does not beg the question, but was remarking that merely thinking about completion as adding elements from some larger ambient space would (without assuming that all Cauchy sequences have a natural limit in the ambient space, which elements do you add?). This conceptual problem doesn't exist when thinking about the completion in terms of equivalence classes of Cauchy sequences, since the Cauchy sequences themselves are in some sense the "added" elements. $\endgroup$ – rem Apr 10 at 16:23
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There is something called the completion of a space, and basically you take as "larger space" the set of Cauchy sequences. You must be a little careful, because you want to identify sequences that "you think should converge to the same thing".This is formalized by saying that two sequences $a_n, b_n$ are the same if the alternating sequence

$$c_{2n} := a_n, c_{2n+1} = b_n$$

is a Cauchy sequence. It is a very instructive exercise to see that the metrics extend to this space, that this space is complete, and that there is a map from the original space to the completion such that the original space is dense.

There is also a faster path to the solutions of the above: google.

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