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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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