دانلود کتاب Concise introduction to basic real analysis

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Cover......Page 1
Half Title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 10
Authors......Page 12
1.1 Introduction and Notations......Page 14
1.3 Relations and Functions......Page 15
1.4 Countable and Uncountable Sets......Page 17
1.5 Set Algebras......Page 19
1.6 Exercises......Page 22
2.1 Field Axioms......Page 26
2.2 Order Axioms......Page 27
2.3 Geometrical Representation of Real Numbers and Intervals......Page 28
2.5 Upper Bounds, Least Upper Bound or Supremum, the Completeness Axiom, Archimedean Property of......Page 29
2.6 Infinite Decimal Representation of Real Numbers......Page 31
2.7 Absolute Value, Triangle Inequality, Cauchy-Schwarz Inequality......Page 33
2.8 Extended Real Number System R*......Page 36
2.9 Exercises......Page 37
3.1 Convergent and Divergent Sequences of Real Numbers......Page 38
3.2 Limit Superior and Limit Inferior of a Sequence of Real Numbers......Page 39
3.3 Infinite Series of Real Numbers......Page 41
3.4 Convergence Tests for Infinite Series......Page 47
3.5 Rearrangements of Series......Page 50
3.6 Riemann\'s Theorem on Conditionally Convergent Series of Real Numbers......Page 51
3.7 Cauchy Multiplications of Series......Page 52
3.8 Exercises......Page 54
4.1 Metric and Metric Spaces......Page 58
4.2 Point Set Topology in Metric Spaces......Page 59
4.3 Convergent and Divergent Sequences in a Metric Space......Page 66
4.4 Cauchy Sequences and Complete Metric Spaces......Page 67
4.5 Exercises......Page 69
5.1 The Limit of Functions......Page 74
5.2 Algebras of Limits......Page 76
5.3 Right-Hand and Left-Hand Limits......Page 79
5.4 Infinite Limits and Limits at Infinity......Page 82
5.5 Certain Important Limits......Page 83
5.6 Sequential Definition of Limit of a Function......Page 84
5.7 Cauchy\'s Criterion for Finite Limits......Page 85
5.8 Monotonic Functions......Page 86
5.10 Continuous and Discontinuous Functions......Page 88
5.11 Some Theorems on the Continuity......Page 93
5.12 Properties of Continuous Functions......Page 96
5.13 Uniform Continuity......Page 98
5.14 Continuity and Uniform Continuity in Metric Spaces......Page 101
5.15 Exercises......Page 104
6.1 Connectedness......Page 112
6.2 The Intermediate Value Theorem......Page 118
6.3 Components......Page 120
6.4 Compactness......Page 121
6.5 The Finite Intersection Property......Page 127
6.6 The Heine-Borel Theorem......Page 129
6.7 Exercises......Page 133
7.1 The Derivative......Page 136
7.2 The Differential Calculus......Page 139
7.3 Properties of Differentiable Functions......Page 145
7.4 The L\'Hospital Rule......Page 151
7.5 Taylor\'s Theorem......Page 160
7.6 Exercises......Page 167
8.1 The Riemann Integral......Page 170
8.2 Properties of the Riemann Integral......Page 181
8.3 The Fundamental Theorems of Calculus......Page 187
8.4 The Substitution Theorem and Integration by Parts......Page 192
8.5 Improper Integrals......Page 194
8.6 The Riemann-Stieltjes Integral......Page 200
8.7 Functions of Bounded Variation......Page 209
8.8 Exercises......Page 218
9.1 The Pointwise Convergence of Sequences of Functions and the Uniform Convergence......Page 226
9.2 The Uniform Convergence and the Continuity, the Cauchy Criterion for the Uniform Convergence......Page 228
9.3 The Uniform Convergence of Infinite Series of Functions......Page 230
9.4 The Uniform Convergence of Integrations and Differentiations......Page 232
9.5 The Equicontinuous Family of Functions and the Arzela-Ascoli Theorem......Page 235
9.6 Dirichlet\'s Test for the Uniform Convergence......Page 237
9.7 The Weierstrass Theorem......Page 238
9.8 Some Examples......Page 240
9.9 Exercises......Page 245
Bibliography......Page 248
Index......Page 250