Page 84 - DMTH401_REAL ANALYSIS

Real Analysis Richa Nandra, Lovely Professional University

Notes Unit 6: Completeness

CONTENTS
Objectives
Introduction
6.1 Completeness
6.2 Proving Cauchy

6.3 Completion
6.4 Contraction Mapping Theorem
6.5 Total Boundedness
6.6 Summary
6.7 Keywords
6.8 Review Questions
6.9 Further Readings

Objectives

After studying this unit, you will be able to:

 Define Completeness
 Discuss the Cauchy
 Explain contraction mapping theorem

 Describe total boundness
Introduction

In earlier unit you have studied about the compactness and connectedness of the set. As you all
come to know about the connected components and Path connectedness. After understanding
the concept of compactness and connectedness let us go through the explanation of completeness.

6.1 Completeness

This is a concept that makes sense in metric spaces only.

Definition: Metric M is complete if every Cauchy sequence in M converges (to a point of M).
Remark: This is not a topological invariant: (0, 1) – incomplete and  complete are homeomorphic.
Proposition: Cvgt  Cauchy.


Proof: "  > 0  N s.t. d(x , x) < for n  N. If m, n  N then
n
2
 
d(x , x ) < d(x , x) + d(x , x) < + = 
m n m n
2 2

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