Banach fixed point theorem

Also known as the contraction mapping theorem, is a fundamental result in the theory of metric spaces, including normed spaces. It provides conditions under which a self-map of a metric space has a unique fixed point.
Theorem
Let $(X,d)$ be a complete metric space and let $f:X\to X$ be a contraction, i.e., there exists a constant $0\le k<1$ such that for all $x,y\in X$, we have $d(f(x),f(y))\le kd(x,y)$. Then $f$ has a unique fixed point, i.e., there exists a unique $x\in X$ such that $f(x)=x$.
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Intuitively, the contraction mapping theorem says that if a map contracts distances in the space, then it has a unique fixed point.