Let be a complete metric space and let be such that is a contraction for some positive integer . Show that has a unique fixed point and that for each , .
Proof. Since is a contraction, then has a fixed point, say , i.e., . Note that
Since then . In other word, is a fixed point of .
To prove the uniqueness, assume $u’_0$ is also a fixed point of . Then both $u_0$ and $u’_0$ are fixed points of which implies that due to the uniqueness of fixed point of a contractive mapping.