## Fixed Point Theorem

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 .

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