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Real Analysis: A First Course / Edition 1
     

Real Analysis: A First Course / Edition 1

by Russell A. Gordon
 

ISBN-10: 0201832100

ISBN-13: 9780201832105

Pub. Date: 07/11/1996

Publisher: Addison-Wesley

This text presents ideas of elementary real analysis, with chapters on real numbers, sequences, limits and continuity, differentiation, integration, infinite series, sequences and series of functions, and point-set topology. Appendices review essential ideas of mathematical logic, sets and functions, and mathematical induction. Students are required to confront formal

Overview

This text presents ideas of elementary real analysis, with chapters on real numbers, sequences, limits and continuity, differentiation, integration, infinite series, sequences and series of functions, and point-set topology. Appendices review essential ideas of mathematical logic, sets and functions, and mathematical induction. Students are required to confront formal proofs. Some background in calculus or linear or abstract algebra is assumed. This second edition adds material on functions of bounded variation, convex functions, numerical methods of integration, and metric spaces. There are 1,600 exercises in this edition, an addition of some 120 pages. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Product Details

ISBN-13:
9780201832105
Publisher:
Addison-Wesley
Publication date:
07/11/1996
Edition description:
Older Edition
Pages:
304
Product dimensions:
6.60(w) x 9.52(h) x 0.71(d)

Table of Contents

1 Real Numbers
1(28)
1.1 What Is a Real Number?
2(6)
1.2 The Completeness Axiom
8(4)
1.3 Some Miscellaneous Results
12(6)
1.4 Countable and Uncountable Sets
18(6)
1.5 The Uncountability of the Set of Real Numbers
24(3)
1.6 Supplementary Exercises
27(2)
2 Sequences
29(18)
2.1 Convergent Sequences
29(7)
2.2 Algebraic Properties and Monotone Sequences
36(4)
2.3 Cauchy Sequences
40(3)
2.4 The Bolzano-Weierstrass Theorem
43(2)
2.5 Supplementary Exercises
45(2)
3 Limits
47(20)
3.1 The Limit of a Function
48(5)
3.2 The Algebra of Limits
53(5)
3.3 One-sided Limits and Infinite Limits
58(5)
3.4 Supplementary Exercises
63(4)
4 Continuity
67(20)
4.1 Continuous Functions
68(4)
4.2 Intermediate and Extreme Values
72(5)
4.3 Monotone Functions and Inverse Functions
77(5)
4.4 Uniform Continuity
82(2)
4.5 Supplementary Exercises
84(3)
5 Differentiation
87(26)
5.1 The Derivative of a Function
88(5)
5.2 The Chain Rule
93(3)
5.3 Further Results on Derivatives
96(5)
5.4 The Mean Value Theorem
101(3)
5.5 L'Hopital's Rule
104(4)
5.6 Supplementary Exercises
108(5)
6 Integration
113(28)
6.1 The Riemann Integral
115(5)
6.2 Properties of the Riemann Integral
120(5)
6.3 Types of Riemann Integrable Functions
125(5)
6.4 The Fundamental Theorem of Calculus
130(3)
6.5 Additional Algebraic Properties
133(4)
6.6 Supplementary Exercises
137(4)
7 Infinite Series
141(28)
7.1 Convergence of Infinite Series
141(7)
7.2 Two Classes of Infinite Series
148(3)
7.3 The Comparison Tests
151(3)
7.4 Absolute Convergence
154(7)
7.5 The Root and Ratio Tests
161(4)
7.6 Supplementary Exercises
165(4)
8 Sequences and Series of Functions
169(40)
8.1 Sequences of Functions
170(4)
8.2 Uniform Convergence
174(3)
8.3 Uniform Convergence and Inherited Properties
177(4)
8.4 Series of Functions
181(4)
8.5 Two Miscellaneous Results
185(6)
8.6 Power Series
191(8)
8.7 Taylor Series
199(6)
8.8 Supplementary Exercises
205(4)
9 Point-Set Topology
209(30)
9.1 Open and Closed Sets
211(5)
9.2 Limit Points
216(2)
9.3 Compact Sets
218(5)
9.4 The Heine-Borel Theorem
223(3)
9.5 Continuous Functions
226(6)
9.6 Continuous Functions and Compact Sets
232(4)
9.7 Supplementary Exercises
236(3)
10 Some Deeper Results
239(20)
10.1 The Set of Points of Continuity of a Function
239(5)
10.2 The Baire Category Theorem
244(4)
10.3 Baire Class One Functions
248(4)
10.4 Continuity Properties of Baire One Functions
252(7)
A Mathematical Logic
259(20)
A.1 Mathematical Theories
259(3)
A.2 Statements and Connectives
262(2)
A.3 Open Statements and Quantifiers
264(3)
A.4 Conditional Statements and Quantifiers
267(3)
A.5 Negation of Quantified Statements
270(2)
A.6 Sample Proofs
272(4)
A.7 Some Words of Advice
276(3)
B Mathematical Induction
279(14)
B.1 Three Equivalent Statements
279(2)
B.2 The Principle of Mathematical Induction
281(5)
B.3 The Principle of Strong Induction
286(3)
B.4 The Well-ordering Property
289(2)
B.5 Some Comments on Induction Arguments
291(2)
References 293(2)
Index of Numbered Items 295(4)
Index 299

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