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This came up in a discussion with Pete L. Clark on this question on complete ordered fields. I argued that every Cauchy sequence in the hyperreal field is eventually constant, hence convergent; he asked whether the same is true for arbitrary Cauchy nets in $\mathbb{R}^*$. I'm not sure how to deduce this either from the transfer principle ("every Cauchy net converges" is a very second-order statement) or from the ultraproduct condition of $\mathbb{R}^*$. Does anyone know the answer?

(I agree that if $f: \mathbb{N}^* \to \mathbb{R}^*$ is an internal Cauchy net, then $f$ has a limit.)

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    If $(x_\lambda)$ is a Cauchy net in a compact space $X$ then it converges to a cluster point $x$. So I guess not every Cauchy net of hyperreals converge since it seems to depend on compactness.2011-03-22

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Hints (general ordered field, not just "the" hyperreal field.)

(a) Can you show that a convergent net is Cauchy?

(b) Are there convergent nets not eventually constant?

(c) Conclude that there are non-constant Cauchy nets.

OF COURSE you need to define "Cauchy net" before you can even ask the question...

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    hmm, the plot thickens. After$I$wrote my last comment, it seems to me that one could at least try to take the "full Cauchy completion" (i.e., the construction using equivalence classes of Cauchy nets) of any non-Archimedean ordered field. I see two potential problems: (i) it may not be completely for free that this gives an ordered field (this cannot be the case for Dedekind completion!). I'm guessing it does though. And (ii): does what one gets deserve to be called a **hyperreal** field? I don't know anything about this...2011-05-13