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Real Analysis - An Introduction
Mathematical Arguments and Elementary Proof Techniques
-
Michael Cullinane
Language:
English
Published/Copyright:
2025
About this book
Designed for a broad spectrum of mathematics majors, not only those pursuing graduate school, this book also provides a thorough explanation of undergraduate Real Analysis. Through a developmentally appropriate narrative that integrates informal discussion, motivation, and basic proof writing approaches with mathematical rigor and clarity, the aim is to assist all students in learning more about the real number system and calculus theory.
- Presents mathematical arguments but also shows how they are developed.
- Makes connections to what students have studied in their calculus courses to motivate interest.
- Includes discussion of elementary mathematical proof techniques.
Author / Editor information
Michael Cullinane, Keene State College, USA.
Supplementary Materials
Topics
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Frontmatter
I -
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Preface
V -
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Contents
VII -
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1 Algebraic and order properties of the real numbers
1 -
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2 Absolute value and distance
23 -
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3 The natural numbers, the integers, and the rational numbers
32 -
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4 The completeness of the real numbers
54 -
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5 Finite sets and infinite sets
70 -
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6 The nested intervals property
86 -
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7 Limit points, closed sets, and open sets
98 -
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8 Connected sets and compact sets
111 -
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9 An introduction to sequences and sequential convergence
121 -
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10 More about sequences and convergence
139 -
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11 Subsequences
158 -
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12 Limits of functions
173 -
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13 An introduction to continuous functions
203 -
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14 More about continuity
218 -
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15 The derivative
237 -
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16 The mean value theorem
252 -
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17 Taylor polynomials
280 -
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18 The Riemann integral
291 -
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19 Criteria for integrability
315 -
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20 The fundamental theorem of calculus
337 -
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21 Approximating the value of an integral
365 -
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22 Infinite series
385 -
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23 Absolute convergence and conditional convergence
409 -
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24 Regroupings and rearrangements of series
429 -
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25 Sequences and series of functions
439 -
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26 Uniform convergence
464 -
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27 More about power series and Taylor series
494 -
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28 The trigonometric functions
516 -
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29 Two remarkable results of Weierstrass
534 -
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A Sets, relations, and functions
543 -
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B Fundamentals of logic
549 -
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C Essential proof strategies
561 -
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D A geometric approach to the cosine and sine functions
565 -
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Bibliography
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Index
573
Publishing information
Pages and Images/Illustrations in book
eBook published on:
August 22, 2025
eBook ISBN:
9783111429564
Paperback published on:
August 22, 2025
Paperback ISBN:
9783111429281
Pages and Images/Illustrations in book
Front matter:
8
Main content:
580
Illustrations:
26
Coloured Illustrations:
54
Tables:
1
Keywords for this book
Theory of calculus; advanced calculus; proofs; real variable theory; real function theory
Audience(s) for this book
Junior and seniors taking an undergraduate course in real analysis.
Safety & product resources
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Manufacturer information:
Walter de Gruyter GmbH
Genthiner Straße 13
10785 Berlin
productsafety@degruyterbrill.com