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Real Analysis                                                   Richa Nandra, Lovely Professional University




                    Notes                            Unit 7: Convergent Sequence


                                     CONTENTS
                                     Objectives
                                     Introduction
                                     7.1  Convergent Sequence
                                     7.2  Properties of Convergent Sequences

                                          7.2.1  Subsequences
                                     7.3  Subsequences and Compact Metric Spaces
                                     7.4  Subsequences Limits
                                     7.5  Cauchy Sequence
                                     7.6  Cauchy Sequence and Closed Sets
                                     7.7  Cauchy Sequences and Convergent Sequences
                                          7.7.1  Complete Spaces

                                     7.8  Increasing/Decreasing Sequences
                                     7.9  Summary
                                     7.10 Keywords
                                     7.11 Review Questions
                                     7.12 Further Readings

                                   Objectives

                                   After studying this unit, you will be able to:
                                      Define convergent sequence

                                      Discuss the properties of convergent sequence
                                      Explain subsequences and compact metric spaces
                                      Describe subsequence limits
                                      Explain the Cauchy sequences and convergent sequences

                                   Introduction

                                   In earlier unit you have studied about the compactness and connectedness of the set. After
                                   understanding the concept of compactness and connectedness let us go through the explanation
                                   of convergent sequence.

                                   7.1 Convergent Sequence


                                   Definition: A sequence {p } in a metric space (X, d) is said to converge if there is a point p  X with
                                                       n
                                   the following property:
                                      ( " > 0)(N) ( " n > N) d(p , p) < 
                                                            n




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