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. A cofinite contraction is a contraction which maps finite sets to cofinite sets. The unique fixed point of every cofinite contraction is infinite. An infinite approximate fixed point is an approximate fixed point whose first set is infinite. Let A be the unique fixed point of the contraction phi.