Banach fixed point theorem

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

Banach Fixed Point Theorem. Every contraction map on a complete metric space has a unique fixed point.