MTH 241 Introduction to Real Analysis

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LINEAR ALGEBRA IN MATHEMATICS

MODULE 1 PRELIMINARIES 1
1.1 Sets and Functions 1
1.2 Mathematical Induction 12
1.3 Finite and Infinite Sets 16

MODULE 2 THE REAL NUMBERS 22
2.1 The Algebraic and Order Properties of R 22
2.2 Absolute Value and Real Line 31
2.3 The Completeness Property of R 34
2.4 Applications of the Supremum Property 38
2.5 Intervals 44

MODULE 3 SEQUENCES AND SERIES 52
3.1 Sequences and Their Limits 53
3.2 Limit Theorems 60
3.3 Monotone Sequences 68
3.4 Subsequences and the Bolzano-Weierstrass Theorem 75
3.5 The Cauchy Criterion 80
3.6 Properly Divergent Sequences 86
3.7 Introduction to Series 89

MODULE 4 LIMITS 96
4.1 Limits of Functions 97
4.2 Limit Theorems 105
4.3 Some Extensions of the Limit Concept 111

MODULE 5 CONTINUOUS FUNCTIONS 119
5.1 Continuous Functions 120
5.2 Combinations of Continuous Functions 125
5.3 Continuous Functions on Intervals 129
5.4 Uniform Continuity 136
5.5 Continuity and Gauges 145
5.6 Monotone and Inverse Functions 149

MODULE 6 DIFFERENTIATION 157
6.1 The Derivative 158
6.2 The Mean Value Theorem 168
6.3 L’Hospital Rules 176
6.4 Taylor’s Theorem 183

MODULE 1 PRELIMINARIES 1

1.1 Sets and Functions 1

1.2 Mathematical Induction 12

1.3 Finite and Infinite Sets 16

MODULE 2 THE REAL NUMBERS 22

2.1 The Algebraic and Order Properties of R 22

2.2 Absolute Value and Real Line 31

2.3 The Completeness Property of R 34

2.4 Applications of the Supremum Property 38

2.5 Intervals 44

MODULE 3 SEQUENCES AND SERIES 52

3.1 Sequences and Their Limits 53

3.2 Limit Theorems 60

3.3 Monotone Sequences 68

3.4 Subsequences and the Bolzano-Weierstrass Theorem 75

3.5 The Cauchy Criterion 80

3.6 Properly Divergent Sequences 86

3.7 Introduction to Series 89

MODULE 4 LIMITS 96

4.1 Limits of Functions 97

4.2 Limit Theorems 105

4.3 Some Extensions of the Limit Concept 111

MODULE 5 CONTINUOUS FUNCTIONS 119

5.1 Continuous Functions 120

5.2 Combinations of Continuous Functions 125

5.3 Continuous Functions on Intervals 129

5.4 Uniform Continuity 136

5.5 Continuity and Gauges 145

5.6 Monotone and Inverse Functions 149

MODULE 6 DIFFERENTIATION 157

6.1 The Derivative 158

6.2 The Mean Value Theorem 168

6.3 L’Hospital Rules 176

6.4 Taylor’s Theorem 183

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MTH 241 Introduction to Real Analysis

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