Theorem (Banach’s fixed point)

Let $(X,d)$ be a non-empty Complete metric space with contraction map $T:X\to X$. Then, $T$ admits a unique fixed point $x^{*}\in X$ with $$T(x^{*})=x^{*}$$Furthermore, for any initial point $x_{0}∈X$ the sequence defined by $x_{n+1} = T(x_n)$ converges to the fixed point $x^*$ as $n \to \infty$.