Sets, Sequences and Mappings: The Basic Concepts of Analysis

Overview


Students progressing to advanced calculus are frequently confounded by the dramatic shift from mechanical to theoretical and from concrete to abstract. This text bridges the gap, offering a systematic development of the real number system and careful treatment of mappings, sequences, limits, continuity, and metric spaces.
The first five chapters consist of a systematic development of many of the important properties of the real number system, plus detailed treatment of such ...
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Sets, Sequences and Mappings: The Basic Concepts of Analysis

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Overview


Students progressing to advanced calculus are frequently confounded by the dramatic shift from mechanical to theoretical and from concrete to abstract. This text bridges the gap, offering a systematic development of the real number system and careful treatment of mappings, sequences, limits, continuity, and metric spaces.
The first five chapters consist of a systematic development of many of the important properties of the real number system, plus detailed treatment of such concepts as mappings, sequences, limits, and continuity. The sixth and final chapter discusses metric spaces and generalizes many of the earlier concepts and results involving arbitrary metric spaces.
An index of axioms and key theorems appears at the end of the book, and more than 300 problems amplify and supplement the material within the text. Geared toward students who have taken several semesters of basic calculus, this volume is an ideal prerequisite for mathematics majors preparing for a two-semester course in advanced calculus. 
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Product Details

  • ISBN-13: 9780486474212
  • Publisher: Dover Publications
  • Publication date: 10/22/2009
  • Series: Dover Books on Mathematics Series
  • Pages: 208
  • Sales rank: 1,367,701
  • Product dimensions: 5.30 (w) x 8.40 (h) x 0.50 (d)

Table of Contents

I Introduction to Sets and Mappings 1

1 Sets 1

2 Mappings 11

3 Real numbers 18

4 Suprema and infima 25

II Sequences 30

1 Monotone and bounded sequences 30

2 Unfirmly isolated sequences 35

III Countable, Connected, Open, and Closed Sets 40

1 Countable sets 40

2 Connected sets. Open intervals and open rays 54

3 Open sets and components 58

4 Closed sets. Compact sets 66

IV Convergence 73

1 Convergent sequences 73

2 Properties of limits 81

3 Cauchy sequences 87

4 Neighborhoods, cluster points, and sequential compactness 90

V Continuity and Uniform Continuity 97

1 Continuous functions 97

2 Other criteria for continuity 113

3 Uniform continuity and extension theorems 121

4 An application of continuous mappings 136

VI Metric Spaces 142

1 Cartesian products, metrics, and metric sets 142

2 Open and closed sets. Metric spaces 147

3 Sequentially compact sets in metric spaces 161

4 Separability and compactness; Lindelösf and Cantor product theorems 169

5 Continuity, uniform continuity, and connectedness 174

Index of Axioms and Key Theorems 185

Index 187

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