The issue of finite forms will be taken up in later postings. We use N for the set of all nonnegative integers. Note that a contraction is determined by its values at finite sets. In this sense, we are in the realm of discrete mathematics. In this context, the classical contraction mapping theorem says the following.
Every contraction has a unique fixed point.
This fixed point may be finite. Let phi be a contraction. Let A be the unique fixed point of the contraction phi. Every decreasing contraction has approximate fixed points of every finite type whose first set consists of even integers. Theorem 5 also holds for the following weakening of Proposition 5.