Basic elements of real analysis

0387984798......Page 1
Contents......Page 10
Preface......Page 8
1.1 Axioms for a Field......Page 14
1.2 Natural Numbers and Sequences......Page 20
1.3 Inequalities......Page 24
1.4 Mathematical Induction......Page 32
2.1 Continuity......Page 36
2.2 Limits......Page 41
2.3 One-Sided Limits......Page 46
2.4 Limits at Infinity; Infinite Limits......Page 50
2.5 Limits of Sequences......Page 55
3.1 The Intermediate-Value Theorem......Page 58
3.2 Least Upper Bound; Greatest Lower Bound......Page 61
3.3 The Bolzano–Weierstrass Theorem......Page 67
3.4 The Boundedness and Extreme-Value Theorems......Page 69
3.5 Uniform Continuity......Page 70
3.6 The Cauchy Criterion......Page 74
3.7 The Heine–Borel Theorem......Page 76
4.1 The Derivative in R[sup(1)]......Page 80
4.2 Inverse Functions in R[sup(1)]......Page 90
5.1 The Darboux Integral for Functions on R[sup(1)]......Page 94
5.2 The Riemann Integral......Page 104
5.3 The Logarithm and Exponential Functions......Page 109
6.1 The Schwarz and Triangle Inequalities; Metric Spaces......Page 114
6.2 Topology in Metric Spaces......Page 119
6.3 Countable and Uncountable Sets......Page 127
6.4 Compact Sets and the Heine–Borel Theorem......Page 131
6.5 Functions on Compact Sets......Page 135
7.1 Partial Derivatives and the Chain Rule......Page 138
7.2 Taylor’s Theorem; Maxima and Minima......Page 143
7.3 The Derivative in R[sup(N)]......Page 149
7.4 The Darboux Integral in R[sup(N)]......Page 154
7.5 The Riemann Integral in R[sup(N)]......Page 158
8.1 Tests for Convergence and Divergence......Page 163
8.2 Series of Positive and Negative Terms; Power Series......Page 168
8.3 Uniform Convergence......Page 175
8.4 Uniform Convergence of Series; Power Series......Page 181
9.1 The Derivative of a Function Defined by an Integral. The Leibniz Rule......Page 191
9.2 Convergence and Divergence of Improper Integrals......Page 196
10.1 Functions of Bounded Variation......Page 203
10.2 The Riemann–Stieltjes Integral......Page 208
11.1 The Implicit Function Theorem......Page 218
11.2 Lagrange Multipliers......Page 223
12.1 Vector Functions on R[sup(N)]......Page 227
12.2 Line Integrals in R[sup(N)]......Page 238
12.3 Green’s Theorem in the Plane......Page 245
12.4 Area of a Surface in R[sup(3)]......Page 252
12.5 The Stokes Theorem......Page 257
12.6 The Divergence Theorem......Page 266
Answers to Odd-Numbered Problems......Page 274
C......Page 282
F......Page 283
L......Page 284
S......Page 285
W......Page 286