One of the interesting problems in the dynamic alternative set theory (AST) is a problem of enlarging or diminishing a class of a finite natural numbers, which we call a horizon. We want to do it in such a way, that a limit of a fixed sequence remains the same with a respect to a new horizon. The most difficult case from this point is a case of p-horizon. In the paper some necessary and sufficient conditions for this possibility are obtained and one counterexample is constructed.