Elementary Real Analysis

PREFACE	xii
VOLUME ONE
1  PROPERTIES OF THE REAL NUMBERS	1
1.1 Introduction	1
1.2 The Real Number System	2
1.3 Algebraic Structure	4
1.4 Order Structure	7
1.5 Bounds	8
1.6 Sups and Infs	9
1.7 The Archimedean Property	12
1.8 Inductive Property of N	13
1.9 The Rational Numbers Are Dense	14
1.10 The Metric Structure of R	16
1.11 Challenging Problems for Chapter 1	18
Notes	20
2 SEQUENCES	21
2.1 Introduction	21
2.2 Sequences	22
2.3 Countable Sets	27
2.4 Convergence	29
2.5 Divergence	33
2.6 Boundedness Properties of Limits	35
2.7 Algebra of Limits	37
2.8 Order Properties of Limits	42
2.9 Monotone Convergence Criterion	47
2.10 Examples of Limits	50
2.11 Subsequences	54
2.12 Cauchy Convergence Criterion	58
2.13 Upper and Lower Limits	61
2.14 Challenging Problems for Chapter 2	66
Notes	69
3 INFINITE SUMS	72
3.1 Introduction	72
3.2 Finite Sums	73
3.3 Infinite Unordered sums	78
3.3.1 Cauchy Criterion	79
3.4 Ordered Sums: Series	83
3.5 Criteria for Convergence	91
3.6 Tests for Convergence	96
3.7 Rearrangements	117
3.8 Products of Series	125
3.9 Summability Methods	128
3.10 More on Infinite Sums	134
3.11 Infinite Products	135
3.12 Challenging Problems for Chapter 3	139
Notes	142
4 SETS OF REAL NUMBERS	146
4.1 Introduction	146
4.2 Points	147
4.3 Sets	153
4.4 Elementary Topology	159
4.5 Compactness Arguments	162
4.6 Countable Sets	173
4.7 Challenging Problems for Chapter 4	174
Notes	177
5 CONTINUOUS FUNCTIONS	179
5.1 Introduction to Limits	179
5.2 Properties of Limits	190
5.3 Limits Superior and Inferior	205
5.4 Continuity	207
5.5 Properties of Continuous Functions	218
5.6 Uniform Continuity	219
5.7 Extremal Properties	222
5.8 Darboux Property	223
5.9 Points of Discontinuity	225
5.10 Challenging Problems for Chapter 5	232
Notes	232
6 MORE ON CONTINUOUS FUNCTIONS AND SETS	239
6.1 Introduction	239
6.2 Dense Sets	239
6.3 Nowhere Dense Sets	241
6.4 The Baire Category Theorem	243
6.5 Cantor Sets	247
6.6 Borel Sets	253
6.7 Oscillation and Continuity	257
6.8 Sets of Measure Zero	263
6.9 Challenging Problems for Chapter 6	268
Notes	268
7 DIFFERENTIATION	271
7.1 Introduction	271
7.2 The Derivative	271
7.3 Computations of Derivatives	278
7.4 Continuity of the Derivative?	288
7.5 Local Extrema	290
7.6 Mean Value Theorem	292
7.7 Monotonicity	298
7.8 Dini Derivates	300
7.9 The Darboux Property of the Derivative	304
7.10 Convexity	307
7.11 L’Hôpital’s Rule	312
7.12 Taylor Polynomials	319
7.13 Challenging Problems for Chapter 7	322
Notes	325
8 THE INTEGRAL	331
8.1 Introduction	331
8.2 Cauchy’s First Method	333
8.3 Properties of the Integral	338
8.4 Cauchy’s Second Method	343
8.5 Cauchy’s Second Method (Continued)	345
8.6 The Riemann Integral	347
8.7 Properties of the Riemann Integral	356
8.8 The Improper Riemann Integral	359
8.9 More on the Fundamental Theorem of Calculus	361
8.10 Challenging Problems for Chapter 8	363
Notes	364
APPENDIX: BACKGROUND	A-1
A.1 Should I Read This Chapter?	A-1
A.2 Notation	A-1
A.3 What Is Analysis?	A-9
A.4 Why Proofs?	A-10
A.5 Indirect Proofhttp://classicalrealanalysis.info/About-us.php	A-11
A.6 Contraposition	A-12
A.7 Counterexamples	A-13
A.8 Induction	A-14
A.9 Quantifiers	A-17
INDEX *	A-20
VOLUME TWO
9 SEQUENCES AND SERIES OF FUNCTIONS	366
9.1 Introduction	366
9.2 Pointwise Limits	367
9.3 Uniform Limits	373
9.4 Uniform Convergence and Continuity	383
9.5 Uniform Convergence and the Integral	387
9.6 Uniform Convergence and Derivatives	393
9.7 Pompeiu’s Function	397
9.8 Continuity and Pointwise Limits	399
9.9 Challenging Problems for Chapter 9	402
Notes	402
10 POWER SERIES	404
10.1 Introduction	404
10.2 Power Series: Convergence	404
10.3 Uniform Convergence	410
10.4 Functions Represented by Power Series	412
10.5 The Taylor Series	418
10.6 Products of Power Series	424
10.7 Composition of Power Series	426
10.8 Trigonometric Series	428
Notes	435
11 THE EUCLIDEAN SPACES Rn	437
11.1 The Algebraic Structure of Rn	437
11.2 The Metric Structure of Rn	439
11.3 Elementary Topology of Rn	442
11.4 Sequences in Rn	445
11.5 Functions and Mappings	448
11.6 Limits of Functions from Rn → Rm	453
11.7 Continuity of Functions from Rn to Rm	458
11.8 Compact Sets in Rn	461
11.9 Continuous Functions on Compact Sets	462
11.10 Additional Remarks	463
Notes	466
12 DIFFERENTIATION ON Rn	467
12.1 Introduction	467
12.2 Partial and Directional Derivatives	467
12.3 Integrals Depending on a Parameter	476
12.4 Differentiable Functions	480
12.5 Chain Rules	494
12.6 Implicit Function Theorems	507
12.7 Functions From R → Rm	520
12.8 Functions From Rn → Rm	523
Notes	535
13 METRIC SPACES	537
13.1 Introduction	537
13.2 Metric Spaces—Specific Examples	539
13.3 Additional Examples	543
13.4 Convergence	548
13.5 Sets in a Metric Space	552
13.6 Functions	558
13.7 Separable Spaces	573
13.8 Complete Spaces	575
13.9 Contraction Maps	581
13.10 Applications of Contraction Maps (I)	588
13.11 Applications of Contraction Maps (II)	591
13.12 Compactness	597
13.13 Baire Category Theorem	614
13.14 Applications of the Baire Category Theorem	620
13.15 Challenging Problems for Chapter 13	627
Notes	630
INDEX *	639

 

 

* The complete Table of Contents and Index are included in each volume.