[PDF] [PDF] Introduction to Real Analysis

[PDF] Stephen Abbott Second Edition - WordPresscom

3 Most importantly, there needs to be significant reward for the difficult The proofs in Understanding Analysis are written with the beginning student firmly in 

[PDF] Abbott Understanding

8 jan 2012 · 3 There needs to be significant reward for the difficult work of firming up the logical structure of limits Specifically, real analysis should not be 

[PDF] Candidate 4 evidence - Understanding Standards

Higher English Reading for Understanding, Analysis and Evaluation 2017 Candidate 4 - evidence SQA www understandingstandards uk 3 of 17 

[PDF] Introduction to Real Analysis

5 jan 2019 · enough that they understand the language, notation, and main ideas And they (iv) The number three, denoted by 3, is the set 2 +

[PDF] MATH 83N: Proofs and Modern Mathematics SYLLABUS Official

Office Hours: Monday 3-4 Office Hours: Tuesday 11-12, Thursday 3-4 Understanding Analysis by Stephen Abbott (2nd edition, ISBN: 1493927116)

[PDF] understanding analysis reddit

[PDF] understanding business intelligence

[PDF] understanding cisco networking technologies

[PDF] understanding cisco networking technologies volume 2 exam 200 301

[PDF] understanding color theory

[PDF] understanding cryptocurrency pdf

[PDF] understanding dental insurance billing

[PDF] understanding dental insurance for providers

[PDF] understanding financial statements

[PDF] understanding ipv4 addressing

[PDF] understanding of art

[PDF] understanding second language acquisition

[PDF] understanding second language acquisition rod ellis free pdf

[PDF] understanding second language acquisition rod ellis pdf download

[PDF] understanding second language acquisition rod ellis pdf free download

Introduction to Real Analysis

Supplementary notes for MATH/MTHE 281

Andrew D. Lewis

This version: 2018/01/09

2

This version: 2018/01/09

Table of Contents

1 Set theory and terminology

1

1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.1.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . .

3

1.1.2 Unions and intersections . . . . . . . . . . . . . . . . . . . . .

5

1.1.3 Finite Cartesian products . . . . . . . . . . . . . . . . . . . . .

7

1.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.2.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . .

12

1.3 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.3.1 Definitions and notation . . . . . . . . . . . . . . . . . . . . .

14

1.3.2 Properties of maps . . . . . . . . . . . . . . . . . . . . . . . . .

16

1.3.3 Graphs and commutative diagrams . . . . . . . . . . . . . . .

19

1.4 Construction of the integers . . . . . . . . . . . . . . . . . . . . . . . .

25

1.4.1 Construction of the natural numbers . . . . . . . . . . . . . .

25

1.4.2 Two relations onZ0. . . . . . . . . . . . . . . . . . . . . . .29

1.4.3 Construction of the integers from the natural numbers . . . .

31

1.4.4 Two relations inZ. . . . . . . . . . . . . . . . . . . . . . . . .34

1.4.5 The absolute value function . . . . . . . . . . . . . . . . . . .

35

1.5 Orders of various sorts . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

1.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

1.5.2 Subsets of partially ordered sets . . . . . . . . . . . . . . . . .

39

1.5.3 Zorn"s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

1.5.4 Induction and recursion . . . . . . . . . . . . . . . . . . . . . .

42

1.5.5 Zermelo"s Well Ordering Theorem . . . . . . . . . . . . . . . .

44

1.5.6 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

1.5.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

1.6 Indexed families of sets and general Cartesian products . . . . . . .

47

1.6.1 Indexed families and multisets . . . . . . . . . . . . . . . . . .

47

1.6.2 General Cartesian products . . . . . . . . . . . . . . . . . . . .

49

1.6.3 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

1.6.4 Directed sets and nets . . . . . . . . . . . . . . . . . . . . . . .

50

1.7 Ordinal numbers, cardinal numbers, cardinality . . . . . . . . . . . .

52

1.7.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . .

52

1.7.2 Cardinal numbers . . . . . . . . . . . . . . . . . . . . . . . . .

56

1.7.3 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

1.8 Some words on axiomatic set theory . . . . . . . . . . . . . . . . . . .

64

1.8.1 Russell"s Paradox . . . . . . . . . . . . . . . . . . . . . . . . .

64

1.8.2 The axioms of Zermelo-Fr

¨ankel set theory . . . . . . . . . . .65

ii

1.8.3 The Axiom of Choice . . . . . . . . . . . . . . . . . . . . . . .

66

1.8.4 Peano"s axioms . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

1.8.5 Discussion of the status of set theory . . . . . . . . . . . . . .

69

1.8.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

1.9 Some words about proving things . . . . . . . . . . . . . . . . . . . .

70

1.9.1 Legitimate proof techniques . . . . . . . . . . . . . . . . . . .

70

1.9.2 Improper proof techniques . . . . . . . . . . . . . . . . . . . .

71

2 Real numbers and their properties

75

2.1 Construction of the real numbers . . . . . . . . . . . . . . . . . . . . .

77

2.1.1 Construction of the rational numbers . . . . . . . . . . . . . .

77

2.1.2 Construction of the real numbers from the rational numbers

82

2.2 Properties of the set of real numbers . . . . . . . . . . . . . . . . . . .

87

2.2.1 Algebraic properties ofR. . . . . . . . . . . . . . . . . . . . .87

2.2.2 The total order onR. . . . . . . . . . . . . . . . . . . . . . . .91

2.2.3 The absolute value function onR. . . . . . . . . . . . . . . .94

2.2.4 Properties ofQas a subset ofR. . . . . . . . . . . . . . . . .95

2.2.5 The extended real line . . . . . . . . . . . . . . . . . . . . . . .

99

2.2.6 sup and inf . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101

2.2.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102

2.3 Sequences inR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104

2.3.1 Definitions and properties of sequences . . . . . . . . . . . .

104

2.3.2 Some properties equivalent to the completeness ofR. . . . .106

2.3.3 Tests for convergence of sequences . . . . . . . . . . . . . . .

109

2.3.4 limsup and liminf . . . . . . . . . . . . . . . . . . . . . . . . .

110

2.3.5 Multiple sequences . . . . . . . . . . . . . . . . . . . . . . . .

113

2.3.6 Algebraic operations on sequences . . . . . . . . . . . . . . .

115

2.3.7 Convergence usingR-nets . . . . . . . . . . . . . . . . . . . .116

2.3.8 A first glimpse of Landau symbols . . . . . . . . . . . . . . .

121

2.3.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123

2.4 Series inR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125

2.4.1 Definitions and properties of series . . . . . . . . . . . . . . .

125

2.4.2 Tests for convergence of series . . . . . . . . . . . . . . . . . .

131

2.4.3 e and. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135

2.4.4 Doubly infinite series . . . . . . . . . . . . . . . . . . . . . . .

139

2.4.5 Multiple series . . . . . . . . . . . . . . . . . . . . . . . . . . .

141

2.4.6 Algebraic operations on series . . . . . . . . . . . . . . . . . .

142

2.4.7 Series with arbitrary index sets . . . . . . . . . . . . . . . . . .

145

2.4.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147

2.5 Subsets ofR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151

2.5.1 Open sets, closed sets, and intervals . . . . . . . . . . . . . . .

151

2.5.2 Partitions of intervals . . . . . . . . . . . . . . . . . . . . . . .

155

2.5.3 Interior, closure, boundary, and related notions . . . . . . . .

156

2.5.4 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . .

162

2.5.5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . .

167
iii

2.5.6 Sets of measure zero . . . . . . . . . . . . . . . . . . . . . . . .

167

2.5.7 Cantor sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171

2.5.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173

3 Functions of a real variable

175

3.1 ContinuousR-valued functions onR. . . . . . . . . . . . . . . . . .178

3.1.1 Definition and properties of continuous functions . . . . . . .

178

3.1.2 Discontinuous functions . . . . . . . . . . . . . . . . . . . . .

182

3.1.3 Continuity and operations on functions . . . . . . . . . . . .

186

3.1.4 Continuity, and compactness and connectedness . . . . . . .

188

3.1.5 Monotonic functions and continuity . . . . . . . . . . . . . . .

191

3.1.6 Convex functions and continuity . . . . . . . . . . . . . . . .

194

3.1.7 Piecewise continuous functions . . . . . . . . . . . . . . . . .

200

3.2 DierentiableR-valued functions onR. . . . . . . . . . . . . . . . .204

3.2.1 Definition of the derivative . . . . . . . . . . . . . . . . . . . .

204

3.2.2 The derivative and continuity . . . . . . . . . . . . . . . . . .

208

3.2.3 The derivative and operations on functions . . . . . . . . . .

211

3.2.4 The derivative and function behaviour . . . . . . . . . . . . .

216

3.2.5 Monotonic functions and dierentiability . . . . . . . . . . .224

3.2.6 Convex functions and dierentiability . . . . . . . . . . . . .231

3.2.7 Piecewise dierentiable functions . . . . . . . . . . . . . . . .237

3.2.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

238

3.3 The Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . . .

240

3.3.1 Step functions . . . . . . . . . . . . . . . . . . . . . . . . . . .

240

3.3.2 The Riemann integral on compact intervals . . . . . . . . . .

242
quotesdbs_dbs17.pdfusesText_23