Are Cauchy sequences natural?

Question

When definining the completion of a field $k$ by a norm one typically uses Cauchy sequences. More specifically the completion of $k$ is defined as the set of equivalence classes of Cauchy sequences in $k$. It seems natural that one would want Cauchy sequences to be convergent in order to say that the field "does not have holes". However I have never seen a justification that would make Cauchy sequences a canonical choice of type of sequence that should be used in the definition of completion. Perhaps there are other equally valid choices to define a completion and they would yield a non-isomorphic field?

Are Cauchy sequences indeed a canonical choice for a definition of the completion? Is there a formulation of the completion by a norm as a universal object in some category or any way to fit Cauchy sequences into a natural setting?

Answer 1

The purpose of defining a Cauchy sequence is that the property is (a) preserved under isometric maps, meaning that a sequence remains Cauchy if you extend or restrict the ambient metric space, and thus (b) is “intrinsic” to the sequence.

Note: By an “isometric map”, I mean a map $f$ from a metric space $(X, d)$ to a metric space $(Y, \rho)$ such that $\rho(f(x_1), f(x_2)) = d(x_1, x_2)$ for all $x_1, x_2 \in X$. An isometric map describes how $(X, d)$ exists as a metric space inside of $(Y, \rho)$.

Here’s a fact you can check, and I hope it answers your question.

Claim: Let $(X, d)$ be a metric space, and let $(x_n)$ be a sequence in $X$. Then the sequence $(x_n)$ is Cauchy in $(X, d)$ iff there exists a metric space $(Y, \rho)$ and an isometric embedding $\iota: X \hookrightarrow Y$ such that the sequence $(\iota(x_n))$ is convergent in $(Y, \rho)$.

The forward direction is just the completion of a metric space. The backward direction follows because if $(\iota(x_n))$ is convergent, then it’s Cauchy, and so $(x_n)$ is also Cauchy.

In other words, a sequence is Cauchy iff there’s some extension of the metric space that makes the sequence convergent. A Cauchy sequence can be made convergent, but a non-Cauchy sequence can never be made convergent by extending the metric space.