- #1

radou

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## Homework Statement

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?