# MTH 241 Introduction to Real Analysis

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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APA

MLA 8th

Ajibade, Samson kay "MTH 241 Introduction to Real Analysis" Afribary. Afribary, 15 Apr. 2020, https://afribary.com/books/mth241-1. Accessed 05 Aug. 2024.

MLA7

Ajibade, Samson kay . "MTH 241 Introduction to Real Analysis". Afribary, Afribary, 15 Apr. 2020. Web. 05 Aug. 2024. < https://afribary.com/books/mth241-1 >.

Chicago

Ajibade, Samson kay . "MTH 241 Introduction to Real Analysis" Afribary (2020). Accessed August 05, 2024. https://afribary.com/books/mth241-1

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