Sets, Sequences and Mappings The Basic Concepts of Analysis

by Anderson, Kenneth; Hall, Dick Wick

• ISBN13: 9780486474212
• ISBN10: 0486474216
• Format: Paperback
• Copyright: 2009-10-22
• Publisher: Dover Publications

Summary

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. 1963 edition.

Author Biography

Both authors taught mathematics at Harpur College, State University of New York, at Binghamton.

Table of Contents

Introduction to Sets and Mappings	p. 1
Sets	p. 1
Mappings	p. 11
Real numbers	p. 18
Suprema and infima	p. 25
Sequences	p. 30
Monotone and bounded sequences	p. 30
Unfirmly isolated sequences	p. 35
Countable, Connected, Open, and Closed Sets	p. 40
Countable sets	p. 40
Connected sets. Open intervals and open rays	p. 54
Open sets and components	p. 58
Closed sets. Compact sets	p. 66
Convergence	p. 73
Convergent sequences	p. 73
Properties of limits	p. 81
Cauchy sequences	p. 87
Neighborhoods, cluster points, and sequential compactness	p. 90
Continuity and Uniform Continuity	p. 97
Continuous functions	p. 97
Other criteria for continuity	p. 113
Uniform continuity and extension theorems	p. 121
An application of continuous mappings	p. 136
Metric Spaces	p. 142
Cartesian products, metrics, and metric sets	p. 142
Open and closed sets. Metric spaces	p. 147
Sequentially compact sets in metric spaces	p. 161
Separability and compactness; Lindelösf and Cantor product theorems	p. 169
Continuity, uniform continuity, and connectedness	p. 174
Index of Axioms and Key Theorems	p. 185
Index	p. 187
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