# Fixed Point Theorem

Submitted by Structure on Sun, 11/11/2007 - 19:09.

Let (X,d) a complete metric space and f:X--->X a contraction (i.e. a function for which there is a real c, such as Then there is a unique such as . Such a point is called a fixed point for f.

Proof.

Unicity. Let two different fixed points.Then . We have

contradiction!

Existence.

Let an arbitrary point and If then is the unique fixed point.

Otherwise, let We shall prove that sequence is a Cauchy sequence and as X is a complete metric space , the sequence will be a convergent sequence.

We have

Then we also have

.

Now let .

if n is big enough as

As any contraction is a continuous function from we have

From

taking limit as p tends to we have also

which gives an estimation of the error made by replacing by .