contraction mapping principle

The contraction mapping principle states that if a function consistently pulls points closer together within a certain space, then applying this function repeatedly will eventually lead to a single, unique point called a fixed point. Think of it like repeatedly folding a paper until it converges at one spot. This principle guarantees that, under these conditions, there is exactly one fixed point, and starting from any initial position, applying the function over and over will bring you closer to that point. It's a foundational idea for ensuring the solutions to certain equations exist and are unique.