I was a bit surprised to find out that one of the exercises in Munkres is actually a proof to the Banach fixed point theorem, unless I'm mistaken. The exercise follows:
If (X, d) is a complete metric space, and f : X --> X a contraction mapping, there is a unique point x0 in X such that f(x0) = x0.
The Attempt at a Solution
Somehow one needs to combine completeness of X with properties of the contraction mapping f, i.e. it is uniformly continuous, for example...
I've beed playing around for some while, trying to follow some ideas, but this seems non-trivial. When I peaked at the proof (I didn't look at it, though), it seemed a bit complicated.
Any discrete hints on solving this one?