Let $(X, d)$ be a [[Metrics|metric space]]. A function $f:X\to X$ is a *contraction* if and only if $d(f(x_{1}), f(x_{2}))\leq \kappa d(x_{1},x_{2}), \quad \forall x_{1},x_{2}\in X$for some $0\leq \kappa<1$. Contractions are examples of Lipschitz continuous functions which are continuous: $x_{n}\to x$ implies $f(x_{n})\to f(x)$. We will use this in the proof of the following theorem which known as *Banach's fixed point theorem* or *Contraction mapping theorem*. # Statements Let $(X,d)$ be a non-empty [[Complete metric spaces|complete metric space]] and $f:X\to X$ a contraction. Then there exists a unique $x\in X$ such that $f(x)=x$ (that is, $f$ has a unique fixed point in $X$). Furthermore, the fixed point $x$ can be found as follows: for any $x_{0} \in X$, the sequence $(x_n)$ defined recursively by $x_{n+1} = f(x_n), \quad n\geq 0,$converges to $x$ as $n \to \infty$. # Proof ###### Sketch of proof First see that if $x,y$ are two fixed points of a contraction $f$, then $x=y$. See that $(x_{n})$ is Cauchy. The completeness of $(X,d)$ then means that it converges to a point $x.$ Since $f$ is continuous, taking the limits of both sides of $x_{n+1}=f(x_{n})$ as $n\to \infty$ yields $x=f(x)$. # Applications See [[Picard–Lindelöf theorem]]. See [[Existence and uniqueness of invariant set of iterated function system]].