**Definition.** Let be a metric space. is said to be complete if every Cauchy sequence in is convergent.

**Definition**. A mapping is called contractive if there exists such that

for all .

**Remark.** If is a contractive, then is continuous.

**Definition.** A point is called a fixed point of if .

**Theorem.** Assume that is a complete metric space, and is contractive. Then has a unique fixed point.

Proof. Take any . For any , we let . Use the assumption of , we have

………. (similarly)

.

Let . Using triangle inequality, we get

as since .

This implies that is a Cauchy sequence. Because is a complete metric space, we conclude that there exists such that .

We claim that . By the continuity of (see Remark), we find that . Take any . We choose such that

for all .

Now we have, keep in mind that ,

.

Since is arbitrary, we get , thus .

For the uniqueness, we assume that for some . Then

Because , we obtain .

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## About baotangquoc

Lecturer
School of Applied Mathematics and Informatics
Hanoi University of Science and Technology
No 1, Dai Co Viet Street, Hanoi

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