The Contraction Mapping Theorem

Prerequisite knowledge and definitions

A metric space is any set of points with some notion of distance between them. Some examples are the real numbers, the integers, or the unit sphere. The distance between these points is given by the metric of the space, hence the name metric space. We will denote the distance between any two points x and y under some metric as d(x,y).

The triangle inequality states that for any three points x y and z in a metric space, d(x,z)≤ d(x,y)+d(y,z).

A cauchy sequence is any sequence of points an where ,for any number ε > 0, there exists a natural number N such that for any natural numbers s and t such that s>N and t>N, d(as,at)<ε. Essentially, what this means is that the terms of the sequence get arbitrarily close together such that at a point an, the entire rest of the sequence stays within a distance of εn from that point, where εn is itself a decreasing sequence of positive numbers that converges to 0.

A complete metric space is a metric space such that every cauchy sequence that it contains converges to some point within that metric space. An example is the real numbers.

A contraction mapping is a function f from a metric space to itself such that for some k < 1, for all points x and y in our metric space, d(f(x),f(y)) ≤ k * d(x,y).

A fixed point of a function f is a point x in a metrix space such that f(x)=x.

Statement of the theorem

Proof of the statement

Cool Consequences