## Basic Real AnalysisOne of the bedrocks of any mathematics education, the study of real analysis introduces students both to mathematical rigor and to the deep theorems and counterexamples that arise from such rigor: for instance, the construction of number systems, the Cantor Set, the Weierstrass nowhere differentiable function, and the Weierstrass approximation theorem. Key features include: * A broad view of mathematics throughout the book * Treatment of all concepts for real numbers first, with extensions to metric spaces later, in a separate chapter * Elegant proofs * Excellent choice of topics * Numerous examples and exercises to enforce methodology; exercises integrated into the main text, as well as at the end of each chapter * Emphasis on monotone functions throughout * Good development of integration theory * Special topics on Banach and Hilbert spaces and Fourier series, often not included in many courses on real analysis * Solid preparation for deeper study of functional analysis * Chapter on elementary probability * Comprehensive bibliography and index * Solutions manual available to instructors upon request By covering all the basics and developing rigor simultaneously, this introduction to real analysis is ideal for senior undergraduates and beginning graduate students, both as a classroom text or for self-study. With its wide range of topics and its view of real analysis in a larger context, the book will be appropriate for more advanced readers as well. |

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### Contents

Set Theory | 1 |

12 Relations and Functions | 5 |

13 Basic Algebra Counting and Arithmetic | 16 |

14 Infinite Direct Products Axiom of Choice and Cardinal Numbers | 28 |

15 Problems | 33 |

Sequences and Series of Real Numbers | 37 |

22 Sequences in ℝ | 47 |

23 Infinite Series | 55 |

72 Some Classes of Integrable Functions | 258 |

73 Sets of Measure Zero and Lebesgues Integrability Criterion | 264 |

74 Properties of the Riemann Integral | 271 |

75 Fundamental Theorem of Calculus | 279 |

76 Functions of Bounded Variation | 283 |

77 Problems | 288 |

Sequences and Series of Functions | 297 |

82 Pointwise and Uniform Convergence | 301 |

24 Unordered Series and Summability | 69 |

25 Problems | 76 |

Limits of Functions | 85 |

31 Bounded and Monotone Functions | 86 |

32 Limits of Functions | 88 |

33 Properties of Limits | 90 |

34 Onesided Limits and Limits Involving Infinity | 94 |

35 Indeterminate Forms Equivalence Landaus Little oh and Big Oh | 102 |

36 Problems | 109 |

Topology of ℝ and Continuity | 113 |

41 Compact and Connected Subsets of ℝ | 114 |

42 The Cantor Set | 118 |

43 Continuous Functions | 122 |

44 Onesided Continuity Discontinuity and Monotonicity | 127 |

45 Extreme Value and Intermediate Value Theorems | 132 |

46 Uniform Continuity | 137 |

47 Approximation by Step Piecewise Linear and Polynomial Functions | 144 |

48 Problems | 150 |

Metric Spaces | 157 |

51 Metrics and Metric Spaces | 158 |

52 Topology of a Metric Space | 162 |

53 Limits Cauchy Sequences and Completeness | 166 |

54 Continuity | 172 |

55 Uniform Continuity and Continuous Extensions | 179 |

56 Compact Metric Spaces | 186 |

57 Connected Metric Spaces | 194 |

58 Problems | 200 |

The Derivative | 209 |

61 Differentiability | 210 |

62 Derivatives of Elementary Functions | 214 |

63 The Differential Calculus | 216 |

64 Mean Value Theorems | 221 |

65 LHôpitals Rule | 226 |

66 Higher Derivatives and Taylors Formula | 230 |

67 Convex Functions | 240 |

68 Problems | 244 |

The Riemann Integral | 251 |

83 Uniform Convergence and Limit Theorems | 307 |

84 Power Series | 312 |

85 Elementary Transcendental Functions | 322 |

86 Fourier Series | 327 |

87 Problems | 343 |

Normed and Function Spaces | 351 |

92 Banach Spaces | 357 |

93 Hilbert Spaces | 366 |

94 Function Spaces | 378 |

95 Problems | 385 |

The Lebesgue Integral F Rieszs Approach | 395 |

101 Improper Riemann Integrals | 396 |

102 Step Functions and Their Integrals | 399 |

103 Convergence Almost Everywhere | 401 |

104 The Lebesgue Integral | 405 |

105 Convergence Theorems | 411 |

106 The Banach Space L¹ | 421 |

107 Problems | 425 |

Lebesgue Measure | 431 |

111 Measurable Functions | 432 |

112 Measurable Sets and Lebesgue Measure | 434 |

113 Measurability Lebesgues Definition | 439 |

114 The Theorems of Egorov Lusin and Steinhaus | 443 |

115 Regularity of Lebesgue Measure | 447 |

116 Lebesgues Outer and Inner Measures | 451 |

117 The Hilbert Spaces L²E 𝔽 | 458 |

118 Problems | 460 |

General Measure and Probability | 467 |

122 Measurable Functions | 481 |

123 Integration | 484 |

124 Probability | 495 |

125 Problems | 513 |

Construction of Real Numbers | 531 |

537 | |

543 | |