Does every Cauchy net of hyperreals converge?
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...