A map is a contraction mapping if it is -Lipschitz with .
Banach Fixed Point Theorem
Let be a contraction mapping in a nonempty complete metric space. Then has a unique fixpoint.
Proof sketch
Define a sequence . This sequence is Cauchy. Hence converges to some . But then also converges to , as is continuous. But also by uniqueness of limits, converges to so .
Suppose . Then
so so , hence unique.