Kumar, Ajit_ Kumaresan, S-A Basic Course in Real Analysis-Taylor & Francis

320 Pages·2012·3.59 MB·English

Statistics
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A BASIC COURSE IN
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REAL ANALYSIS EA RREEAALL
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Based on the authors’ combined 35 years of experience in teaching, I
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A Basic Course in Real Analysis introduces students to the aspects

of real analysis in a friendly way. The authors offer insights into the NC
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way a typical mathematician works observing patterns, conducting
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experiments by means of looking at or creating examples, trying to
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understand the underlying principles, and coming up with guesses
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or conjectures and then proving them rigorously based on his or her YE

explorations. SI
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With more than 100 pictures, the book creates interest in real analysis I
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by encouraging students to think geometrically. Each difficult proof
is prefaced by a strategy and explanation of how the strategy is
translated into rigorous and precise proofs. The authors then explain
the mystery and role of inequalities in analysis to train students to
arrive at estimates that will be useful for proofs. They highlight the role K
of the least upper bound property of real numbers, which underlies U
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all crucial results in real analysis. In addition, the book demonstrates
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analysis as a qualitative as well as quantitative study of functions,
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exposing students to arguments that fall under hard analysis.
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Although there are many books available on this subject, students U
often find it difficult to learn the essence of analysis on their own or after M
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going through a course on real analysis. Written in a conversational
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tone, this book explains the hows and whys of real analysis and
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provides guidance that makes readers think at every stage. S
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A BASIC COURSE IN
REAL
ANALYSIS

A BASIC COURSE IN
REAL
ANALYSIS
AJIT KUMAR
S. KUMARESAN
CRC Press
Taylor & Francis Group
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Version Date: 20130911
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Dedicated to
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and especially to the
Mathematics Training and Talent Search Programme, India

Contents
Preface xi
To the Students xiii
About the Authors xv
List of Figures xvii
1 Real Number System 1
1.1 Algebra of the Real Number System . . . . . . . . . . . . . . . . . 1
1.2 Upper and Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 LUB Property and Its Applications . . . . . . . . . . . . . . . . . . 7
1.4 Absolute Value and Triangle Inequality . . . . . . . . . . . . . . . 20
2 Sequences and Their Convergence 27
2.1 Sequences and Their Convergence. . . . . . . . . . . . . . . . . . . 28
2.2 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Monotone Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4 Sandwich Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5 Some Important Limits . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Sequences Diverging to ±∞ . . . . . . . . . . . . . . . . . . . . . . 52
2.7 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.8 Sequences Defined Recursively . . . . . . . . . . . . . . . . . . . . 58
3 Continuity 63
3.1 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 ε-δ Definition of Continuity . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . . 78
3.4 Extreme Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . 84
3.5 Monotone Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.6 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.7 Uniform Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.8 Continuous Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 103
vii
viii CONTENTS
4 Differentiation 109
4.1 Differentiability of Functions . . . . . . . . . . . . . . . . . . . . . 110
4.2 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.3 L’Hospital’s Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.4 Higher-order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 134
4.5 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.6 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.7 Cauchy’s Form of the Remainder . . . . . . . . . . . . . . . . . . . 150
5 Infinite Series 153
5.1 Convergence of an Infinite Series . . . . . . . . . . . . . . . . . . . 154
5.2 Abel’s Summation by Parts . . . . . . . . . . . . . . . . . . . . . . 163
5.3 Rearrangements of an Infinite Series . . . . . . . . . . . . . . . . . 165
5.4 Cauchy Product of Two Infinite Series . . . . . . . . . . . . . . . . 172
6 Riemann Integration 175
6.1 Darboux Integrability . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.2 Properties of the Integral . . . . . . . . . . . . . . . . . . . . . . . 186
6.3 Fundamental Theorems of Calculus . . . . . . . . . . . . . . . . . . 194
6.4 Mean Value Theorems for Integrals . . . . . . . . . . . . . . . . . . 199
6.5 Integral Form of the Remainder . . . . . . . . . . . . . . . . . . . . 203
6.6 Riemann’s Original Definition . . . . . . . . . . . . . . . . . . . . . 205
6.7 Sum of an Infinite Series as an Integral . . . . . . . . . . . . . . . . 210
6.8 Logarithmic and Exponential Functions . . . . . . . . . . . . . . . 212
6.9 Improper Riemann Integrals . . . . . . . . . . . . . . . . . . . . . . 214
7 Sequences and Series of Functions 221
7.1 Pointwise Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 221
7.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 228
7.3 Consequences of Uniform Convergence . . . . . . . . . . . . . . . . 231
7.4 Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.5 Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
7.6 Taylor Series of a Smooth Function . . . . . . . . . . . . . . . . . . 258
7.7 Binomial Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
7.8 Weierstrass Approximation Theorem . . . . . . . . . . . . . . . . . 264
A Quantifiers 271
B Limit Inferior and Limit Superior 277
C Topics for Student Seminars 283
CONTENTS ix
D Hints for Selected Exercises 287
D.1 Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
D.2 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
D.3 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
D.4 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
D.5 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
D.6 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
D.7 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Bibliography 297
Index 299