Banach fixed-point theorem

The Banach fixed-point theorem is the workhorse that turns repeated approximation into a guarantee. If a map squeezes a complete space by a uniform factor, then it has one and only one point it leaves fixed, and you can find that point by starting anywhere and applying the map over and over.

Precisely: let (X, d) be a non-empty complete metric space and let T: X -> X be a contraction with factor k < 1. Then T has a unique fixed point x-star, meaning T(x-star) = x-star. Moreover, for any starting point x_0, the iterates x_{n+1} = T(x_n) converge to x-star, with the explicit error bound d(x_n, x-star) <= k^n / (1 - k) times d(x_1, x_0).

All three hypotheses earn their keep: completeness ensures the Cauchy sequence of iterates actually has a limit, the strict bound k < 1 forces the iterates to be Cauchy and the fixed point to be unique, and T must map X into itself. Drop completeness, or allow k = 1, and uniqueness or existence can fail. The theorem is the engine behind existence-and-uniqueness proofs for differential equations (Picard's theorem) and for the inverse and implicit function theorems.