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.)
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...