Do all Cauchy sequences in incomplete metric spaces have "natural limits" that simply lie outside the space?

Question

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.)

Answer 1

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.

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