Contraction Mapping Theorem

The Contraction Mapping Theorem states that in a complete space, a function that always brings points closer together (a contraction) has exactly one fixed point, meaning applying the function to that point leaves it unchanged. More practically, if you repeatedly apply such a function starting from any point, you'll eventually get closer and closer to this fixed point. This guarantees both the existence and uniqueness of the solution and provides a method for finding it through iteration. It's fundamental in analysis because it ensures that certain equations have unique solutions that can be found reliably.