**Pure and Applied Undergraduate Texts**

Volume: 25;
2016;
348 pp;
Hardcover

MSC: Primary 26; 54;

**Print ISBN: 978-1-4704-2807-5
Product Code: AMSTEXT/25**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

**Electronic ISBN: 978-1-4704-3494-6
Product Code: AMSTEXT/25.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

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#### Supplemental Materials

# Mathematical Analysis and Its Inherent Nature

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*Hossein Hosseini Giv*

Mathematical analysis is often referred to as generalized
calculus. But it is much more than that. This book has been written in
the belief that emphasizing the inherent nature of a mathematical
discipline helps students to understand it better. With this in mind,
and focusing on the essence of analysis, the text is divided into two
parts based on the way they are related to calculus: completion and
abstraction. The first part describes those aspects of analysis which
complete a corresponding area of calculus theoretically, while the
second part concentrates on the way analysis generalizes some aspects
of calculus to a more general framework. Presenting the contents in
this way has an important advantage: students first learn the most
important aspects of analysis on the classical space \(\mathbb{R}\) and
fill in the gaps of their calculus-based knowledge. Then they proceed
to a step-by-step development of an abstract theory, namely, the
theory of metric spaces which studies such crucial notions as limit,
continuity, and convergence in a wider context.

The readers are assumed to have passed courses in one- and
several-variable calculus and an elementary course on the foundations
of mathematics. A large variety of exercises and the inclusion of
informal interpretations of many results and examples will greatly
facilitate the reader's study of the subject.

#### Readership

Undergraduate students interested in mathematical analysis.

#### Reviews & Endorsements

[T]his is an attractive text, one that certainly merits a look by anybody trying to find a text for a course in undergraduate analysis.

-- Mark Hunacek, MAA Reviews

#### Table of Contents

# Table of Contents

## Mathematical Analysis and Its Inherent Nature

- Cover Cover11
- Title page iii4
- Contents vii8
- To the Instructor xi12
- To the Student xiii14
- Introduction and Outline of the Book xvii18
- Acknowledgments xxi22
- Part 1 . Rebuilding the Calculus Building 124
- Chapter 1. The Real Number System Revisited 326
- 1.1. The Algebraic Axioms 528
- 1.2. The Order Axioms 730
- 1.3. Absolute Value, Distance, and Neighborhoods 932
- 1.4. Natural Numbers and Mathematical Induction 1235
- 1.5. The Axiom of Completeness and Its Uses 2144
- 1.6. The Complex Number System 3356
- Notes on Essence and Generalizability 3962
- Exercises 4164

- Chapter 2. Sequences and Series of Real Numbers 4770
- 2.1. Real Sequences, Their Convergence, and Boundedness 4871
- 2.2. Subsequences, Limit Superior and Limit Inferior 6689
- 2.3. Cauchy Sequences 7497
- 2.4. Sequences in Closed and Bounded Intervals 7699
- 2.5. Series: Revisiting Some Convergence Tests 78101
- 2.6. Rearrangements of Series 90113
- 2.7. Power Series 92115
- Notes on Essence and Generalizability 96119
- Exercises 97120

- Chapter 3. Limit and Continuity of Real Functions 103126
- 3.1. Limit Points and Some Other Classes of Points in ℝ 104127
- 3.2. A More General Definition of Limit 112135
- 3.3. Limit at Infinity 126149
- 3.4. One-Sided Limits\index{one-sided!limit} 130153
- 3.5. Continuity and Two Kinds of Discontinuity 136159
- 3.6. Continuity on [𝑎,𝑏]: Results and Applications 142165
- 3.7. Uniform Continuity 149172
- Notes on Essence and Generalizability 151174
- Exercises 152175

- Chapter 4. Derivative and Differentiation 159182
- 4.1. The Why and What of the Concept of Derivative 160183
- 4.2. The Basic Properties of Derivative 168191
- 4.3. Local Extrema and Derivative 172195
- 4.4. The Mean Value Theorem: More Applications of Derivative 175198
- 4.5. Taylor Series: A First Glance 181204
- 4.6. Taylor’s Theorem and the Convergence of Taylor Series 185208
- Notes on Essence and Generalizability 189212
- Exercises 190213

- Chapter 5. The Riemann Integral 193216

- Part 2 . Abstraction and Generalization 233256
- Chapter 6. Basic Theory of Metric Spaces 235258
- Chapter 7. Sequences in General Metric Spaces 275298
- 7.1. Convergence and Divergence in Metric Spaces 275298
- 7.2. Cauchy Sequences and Complete Metric Spaces 284307
- 7.3. Compactness: Definition and Some Basic Results 287310
- 7.4. Compactness: Some Equivalent Forms 290313
- 7.5. Perfect Sets and Cantor’s Set 294317
- Notes on Essence and Generalizability 296319
- Exercises 296319

- Chapter 8. Limit and Continuity of Functions in Metric Spaces 299322
- 8.1. The Definition of Limit in General Metric Spaces 299322
- 8.2. Continuity and Uniform Continuity 302325
- 8.3. Continuity and Compactness 307330
- 8.4. Connectedness and Its Relation to Continuity 310333
- 8.5. Banach’s Fixed Point Theorem 314337
- Notes on Essence and Generalizability 316339
- Exercises 317340

- Chapter 9. Sequences and Series of Functions 319342
- Appendix 337360
- Bibliography 343366
- Index 345368

- Back Cover Back Cover1374