Two theorems on finite unions of regressive immune sets

It is proved that the set of all natural numbers cannot be represented as the union of a finite number of regressive immune sets. This answers a question of Appel and McLaughlin. Incidentally, we obtain the following two results: 1. If A₁, ..., A_n are regressive immune sets, then there exists a general recursive function f such that D_f(0), ..., D_f(n), ... is a sequence of pairwise disjoint sets and $$\forall ^x (|D_{f(x)} |) \leqslant n + 1\& D_{f(x)} \cap \overline {A_1 \cup ... \cup A_n } \ne \emptyset )$$ .2. If A₁, ..., A_n are regressive and B is an infinite subset of \(\bigcup\limits_{t = 1}^n {A_1 } \) , then there exists an i that A_i?_eB.