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E-raamat: Real Analysis: With Proof Strategies

  • Formaat: 281 pages
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 19-Jan-2021
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-13: 9781000294248
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Real Analysis: With Proof StrategiesSuurem pilt
  • Formaat: 281 pages
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 19-Jan-2021
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-13: 9781000294248
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This book provides a resolution to the "bridging-the-gap problem." The book not only presents the fundamental theorems of real analysis, but also shows the reader how to compose and produce the proofs of these theorems.

 



Typically, undergraduates see real analysis as one of the most di cult courses that a mathematics major is required to take. The main reason for this perception is twofold: Students must comprehend new abstract concepts and learn to deal with these concepts on a level of rigor and proof not previously encountered. A key challenge for an instructor of real analysis is to ?nd a way to bridge the gap between a student’s preparation and the mathematical skills that are required to be successful in such a course.

Real Analysis: With Proof Strategies

provides a resolution to the "bridging-the-gap problem." The book not only presents the fundamental theorems of real analysis, but also shows the reader how to compose and produce the proofs of these theorems. The detail, rigor, and proof strategies o ered in this textbook will be appreciated by all readers.

Features

  • Explicitly shows the reader how to produce and compose the proofs of the basic theorems in real analysis

  • Suitable for junior or senior undergraduates majoring in mathematics.

Arvustused

"This textbook is intended for undergraduate students who have completed a standard calculus course sequence that covers differentiation and integration and a course that introduces the basics of proof-writing. For students who have a limited proof-writing background, the author includes an abbreviated discussion of proofs, sets, functions, and induction in Chapter 1.

[ . . . ] In summary, this book is a good resource for students who are taking a first course in real analysis and who have a limited background in proof-writing." MAA Reviews

"Real Analysis: With Proof Strategies by Professor Daniel W. Cunningham explicitly shows the reader how to produce and compose the proofs of the basic theorems in real analysis and is eminently suitable for junior or senior undergraduates majoring in mathematics." Midwest Book Review

Preface ix
Chapter 1 Proofs, Sets, Functions, and Induction
1(28)
1.1 PROOFS
1(9)
1.1.1 Important Sets in Mathematics
1(2)
1.1.2 How to Prove an Equation
3(1)
1.1.3 How to Prove an Inequality
4(2)
1.1.4 Important Properties of Absolute Value
6(1)
1.1.5 Proof Diagrams
7(3)
1.2 SETS
10(5)
1.2.1 Basic Definitions of Set Theory
10(1)
1.2.2 Set Operations
10(1)
1.2.3 Indexed Families of Sets
11(1)
1.2.4 Generalized Unions and Intersections
12(2)
1.2.5 Unindexed Families of Sets
14(1)
1.3 FUNCTIONS
15(8)
1.3.1 Real-Valued Functions
16(1)
1.3.2 Injections and Surjections
16(1)
1.3.3 Composition of Functions
17(1)
1.3.4 Inverse Functions
18(1)
1.3.5 Functions Acting on Sets
19(4)
1.4 MATHEMATICAL INDUCTION
23(6)
1.4.1 The Well-Ordering Principle
23(1)
1.4.2 Proof by Mathematical Induction
23(6)
Chapter 2 The Real Numbers
29(26)
2.1 Introduction
29(1)
2.2 R Is An Ordered Field
30(6)
2.2.1 The Absolute Value Function
33(3)
2.3 The Completeness Axiom
36(11)
2.3.1 Proofs on the Supremum of a Set
39(1)
2.3.2 Proofs on the Infimum of a Set
40(3)
2.3.3 Alternative Proof Strategies
43(4)
2.4 The Archimedean Property
47(3)
2.4.1 The Density of the Rational Numbers
48(2)
2.5 Nested Intervals Theorem
50(5)
2.5.1 E is Uncountable
51(4)
Chapter 3 Sequences
55(46)
3.1 Convergence
56(12)
3.2 Limit Theorems For Sequences
68(7)
3.2.1 Algebraic Limit Theorems
68(4)
3.2.2 The Squeeze Theorem
72(1)
3.2.3 Order Limit Theorems
73(2)
3.3 Subsequences
75(4)
3.4 Monotone Sequences
79(6)
3.4.1 The Monotone Subsequence Theorem
83(2)
3.5 Bolzano-Weierstrass Theorems
85(2)
3.6 Cauchy Sequences
87(4)
3.7 Infinite Limits
91(2)
3.8 Limit Superior and Limit Inferior
93(8)
3.8.1 The Limit Superior of a Bounded Sequence
93(3)
3.8.2 The Limit Inferior of a Bounded Sequence
96(1)
3.8.3 Connections and Relations
97(4)
Chapter 4 Continuity
101(28)
4.1 Continuous Functions
101(9)
4.2 Continuity and Sequences
110(2)
4.3 Limits of Functions
112(7)
4.4 Consequences of Continuity
119(5)
4.4.1 The Extreme Value Theorem
119(1)
4.4.2 The Intermediate Value Theorem
120(4)
4.5 Uniform Continuity
124(5)
Chapter 5 Differentiation
129(22)
5.1 The Derivative
129(6)
5.1.1 The Rules of Differentiation
131(2)
5.1.2 The Chain Rule
133(2)
5.2 The Mean Value Theorem
135(12)
5.2.1 L'Hopital's Rule
139(4)
5.2.2 The Intermediate Value Theorem for Derivatives
143(1)
5.2.3 Inverse Function Theorems
144(3)
5.3 Taylor's Theorem
147(4)
Chapter 6 Riemann Integration
151(38)
6.1 The Riemann Integral
151(11)
6.1.1 Partitions and Darboux Sums
152(1)
6.1.2 Basic Results Regarding Darboux Sums
153(3)
6.1.3 The Definition of the Riemann Integral
156(2)
6.1.4 A Necessary and Sufficient Condition
158(4)
6.2 Properties of the Riemann Integral
162(8)
6.2.1 Linearity Properties
162(3)
6.2.2 Oder Properties
165(1)
6.2.3 Integration over Subintervals
166(2)
6.2.4 The Composition Theorem
168(2)
6.3 Families of Integrable Functions
170(7)
6.3.1 Continuous Functions
170(2)
6.3.2 Monotone Functions
172(1)
6.3.3 Functions of Bounded Variation
172(5)
6.4 The Fundamental Theorem of Calculus
177(12)
6.4.1 Evaluating Riemann Integrals
177(1)
6.4.2 Continuous Functions have Antiderivatives
178(2)
6.4.3 Techniques of Antidifferentiation
180(4)
6.4.4 Improper Integrals
184(5)
Chapter 7 Infinite Series
189(24)
7.1 Convergence and Divergence
189(7)
7.2 Convergence Tests
196(9)
7.2.1 Comparison Tests
196(2)
7.2.2 The Integral Test
198(1)
7.2.3 Alternating Series
199(1)
7.2.4 Absolute Convergence
200(1)
7.2.5 The Ratio and Root Tests
201(4)
7.3 Regrouping and Rearranging Terms of a Series
205(8)
7.3.1 Regrouping
206(1)
7.3.2 Rearrangements
207(6)
Chapter 8 Sequences and Series of Functions
213(22)
8.1 Pointwise and Uniform Convergence
213(5)
8.1.1 Sequences of Functions
213(3)
8.1.2 Series of Functions
216(2)
8.2 Preservation Theorems
218(6)
8.3 Power Series
224(6)
8.4 Taylor Series
230(5)
Appendix A Proof of the Composition Theorem
235(4)
Appendix B Topology on the Real Numbers
239(14)
B.1 Open and Closed Sets
239(5)
B.1.1 Continuity Revisited
241(1)
B.1.2 Accumulation Points Revisited
242(2)
B.2 Compact Sets
244(3)
B.3 The Heine-Borel Theorem
247(6)
B.3.1 The Finite Intersection Property
249(1)
B.3.2 The Cantor Set
249(4)
Appendix C Review of Proof and Logic
253(8)
Bibliography 261(2)
List of Symbols 263(2)
Index 265
Daniel W. Cunningham is a Professor of Mathematics at SUNY Buffalo State, a campus of the State University of New York. He was born and raised in Southern California and holds a Ph.D. in mathematics from the University of California at Los Angeles (UCLA). He is also a member of the Association for Symbolic Logic, the American Mathematical Society, and the Mathematical Association of America.

Cunningham is the author of multiple books. Before arriving at Buffalo State, Professor Cunningham worked as a software engineer in the aerospace industry
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