Contracting Mapping Theorem

Statements

Theorem. Let $(X,d)$ me a complete metric space and let $T:M\to M$. If there exists a constant $\lambda \in [0,1)$ such that, for every $x_1,x_2\in X$ the inequality $d(T(x_1),T(x_2))\le \lambda d(x_1,x_2)$ holds, then $T$ has a unique fixed point $a$ in $M$.

References

@Boo75 Theorem 6.5