A First Course in Real Analysis, Second Edition

Cover......Page 1
Title......Page 4
Preface to the Second Edition......Page 8
Preface to the First Edition......Page 12
Contents......Page 16
1.1 Axioms for a Field......Page 20
1.2 Natural Numbers and Sequences......Page 28
1.3 Inequalities......Page 34
1.4 Mathematical Induction......Page 44
2.1 Continuity......Page 49
2.2 Limits......Page 54
2.3 One-Sided Limits......Page 61
2.4 Limits at Infinity; Infinite Limits......Page 67
2.5 Limits of Sequences......Page 74
3.1 The Intermediate-Value Theorem......Page 78
3.2 Least Upper Bound; Greatest Lower Bound......Page 81
3.3 The Bolzano-Weierstrass Theorem......Page 87
3.4 The Boundedness and Extreme-Value Theorems......Page 89
3.5 Uniform Continuity......Page 91
3.6 The Cauchy Criterion......Page 94
3.7 The Heine-Bore! and Lebesgue Theorems......Page 96
4.1 The Derivative in ~ 1......Page 102
4.2 Inverse Functions in ~ 1......Page 113
5.1 The Darboux Integral for Functions on ~ 1......Page 117
5.2 The Riemann Integral......Page 130
5.3 The Logarithm and Exponential Functions......Page 136
5.4 Jordan Content and Area......Page 141
6.1 The Schwarz and Triangle Inequalities; Metric Spaces......Page 149
6.2 Elements of Point Set Topology......Page 155
6.3 Countable and Uncountable Sets......Page 164
6.4 Compact Sets and the Heine-Borel Theorem......Page 169
6.5 Functions on Compact Sets......Page 176
6.6 Connected Sets......Page 180
6.7 Mappings from One Metric Space to Another......Page 183
7.1 Partial Derivatives and the Chain Rule......Page 192
7.2 Taylor\'s Theorem; Maxima and Minima......Page 197
7.3 The Derivative in ~N......Page 207
8.1 Volume in ~N......Page 213
8.2 The Darboux Integral in ~N......Page 216
8.3 The Riemann Integral in ~N......Page 222
9.1 Tests for Convergence and Divergence......Page 230
9.2 Series of Positive and Negative Terms; Power Series......Page 235
9.3 Uniform Convergence of Sequences......Page 241
9.4 Uniform Convergence of Series; Power Series......Page 249
9.5 Unordered Sums......Page 260
9.6 The Comparison Test for Unordered Sums; Uniform Convergence......Page 269
9.7 Multiple Sequences and Series......Page 273
10.1 Expansions of Periodic Functions......Page 282
10.2 Sine Series and Cosine Series; Change oflnterval......Page 289
10.3 Convergence Theorems......Page 294
11.1 The Derivative of a Function Defined by an Integral; the Leibniz Rule......Page 304
11.2 Convergence and Divergence of Improper Integrals......Page 309
11.3 The Derivative of Functions Defined by Improper Integrals; the Gamma Function......Page 314
12.1 Functions of Bounded Variation......Page 324
12.2 The Riemann-Stieltjes Integral......Page 335
13.1 A Fixed Point Theorem and Newton\'s Method......Page 348
13.2 Application of the Fixed Point Theorem to Differential Equations......Page 354
14.1 The Implicit Function Theorem for a Single Equation......Page 360
14.2 The Implicit Function Theorem for Systems......Page 367
14.3 Change of Variables in a Multiple Integral......Page 378
14.4 The Lagrange Multiplier Rule......Page 388
15.1 Complete Metric Spaces......Page 393
15.2 Convex Sets and Convex Functions......Page 400
15.3 Arzela\'s Theorem; the Tietze Extension Theorem......Page 412
15.4 Approximations and the Stone-Weierstrass Theorem......Page 422
16.1 Vector Functions on R1......Page 432
16.2 Vector Functions and Fields on RN......Page 442
16.3 Line Integrals in RN......Page 453
16.4 Green\'s Theorem in the Plane......Page 464
16.5 Surfaces in R3 ; Parametric Representation......Page 474
16.6 Area of a Surface in R3 ; Surface Integrals......Page 480
16.7 Orientable Surfaces......Page 490
16.8 The Stokes Theorem......Page 496
16.9 The Divergence Theorem......Page 505
Appendix 1 Absolute Value......Page 514
Appendix 2 Solution of Algebraic Inequalities......Page 518
Appendix 3 Expansions of Real Numbers in Any Base......Page 522
Appendix 4 Vectors in EN......Page 526
Answers to Odd-Numbered Problems......Page 534
Index......Page 548