Contents

Preface

1. Real Numbers and Monotone Sequences 1
1.1 Introduction; Real numbers 1
1.2 Increasing sequences 3
1.3 Limit of an increasing sequence 4
1.4 Example: the number e 5
1.5 Example: the harmonic sum and Euler's number, 8
1.6 Decreasing sequences; Completeness property 10
2. Estimations and Approximations 17
2.1 Introduction; Inequalities 17
2.2 Estimations 18
2.3 Proving boundedness 20
2.4 Absolute values; estimating size 21
2. 5 Approximations 24
2.6 The terminology "for n large" 27
3. The Limit of a Sequence 35
3.1 Definition of limit 35
3.2 Uniqueness of limits; the K-e principle 38
3.3 Infinite limits 40
3.4 Limit of an 42
3.5 Writing limit proofs 43
3.6 Some limits involving integrals 44
3.7 Another limit involving an integral 45
4. The Error Term 51
4.1 The error term 51
4.2 The error in the geometric series; Applications 52
4.3 A sequence converging to J2: Newton's method 53
4.4 The sequence of Fibonacci fractions 56
5. Limit Theorems for Sequences 61
5.1 Limits of sums, products, and quotients 61
5.2 Comparison theorems 64
5.3 Location theorems 67
5.4 Subsequences; Non-existence of limits 68
5.5 Two common mistakes 71

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Introduction to Analysis

6. The Completeness Property 78
6.1 Introduction; Nested intervals 78
6.2 Cluster points of sequences 80
6.3 The Bolzano- Weierstrass theorem 82
6.4 Cauchy sequences 83
6.5 Completeness property for sets 86

7. Infinite Series 94
7.1 Series and sequences 94
7.2 Elementary convergence tests 97
7.3 The convergence of series with negative terms 100
7.4 Convergence tests: ratio and n-th root tests 102
7.5 The integral and asymptotic comparison tests 104
7.6 Series with alternating signs: Cauchy's test 106
7.7 Rearranging the terms of a series 107

8. Power Series 114
8.1 Introduction; Radius of convergence 114
8.2 Convergence at the endpoints; Abel summation 117
8.3 Operations on power series: addition 119
8.4 Multiplication of power series 120

9. Functions of One Variable 125
9.1 Functions 125
9.2 Algebraic operations on functions 127
9.3 Some properties of functions 128
9.4 Inverse functions 131
9.5 The elementary functions 133

10. Local and Global Behavior 137
10.1 Intervals; estimating functions 137
10.2 Approximating functions 141
10.3 Local behavior 143
10.4 Local and global properties of functions 145

11. Continuity and Limits of Functions 151
11.1 Continuous functions 151
11.2 Limits of functions 155
11.3 Limit theorems for functions 158
11.4 Limits and continuous functions 162
11.5 Continuity and sequences 155

12. The Intermediate Value Theorem 172
12.1 The existence of zeros 172
12.2 Applications of Bolzano's theorem 175
12.3 Graphical continuity 178
12.4 Inverse functions 179