The work is carried out in the alternative set theory. We study conditions on a cut of a class of natural numbers, provided that hyperreal structure based on the cut has Cauchy sequential property. It is proved, that the necessary and sufficient condition for it is additivity of the cut. We show also that corresponding structure contains all limits of its monotone bounded sequences if and only if the cut is closed with a respect to any growing set function.

Keywords: alternative set theory, cuts, Cauchy sequence.