Let $(X,d)$ be a complete metric space. A function $T:X \to X$ is said to be a contraction mapping if there is a constant $q$ with $0 \leq q < 1$ such that $$ d(Tx,Ty)\leq q\cdot d(x,y) $$ for all $x,y\in X$ . Contractions have an important property.

There is an estimate to this fixed point that can be useful in applications. Let $T$ be a contraction mapping on $(X,d)$ with constant $q$ and unique fixed point $x^* \in X$ . For any $x_0 \in X$ , define recursively the following sequence \begin{eqnarray*} x_1 &:=& Tx_0 \\ x_2 &:=& Tx_1 \\ &\vdots& \\ x_{n+1} &:=& Tx_n. \end{eqnarray*}The following inequality then holds: $$ d(x^*,x_n)\leq \frac{q^n}{1-q}d(x_1,x_0). $$ So the sequence $(x_n)$ converges to $x^*$ . This recursive estimate is occasionally responsible for this result being known as the method of successive approximations.