**AMS/MAA Textbooks**

Volume: 31;
2015;
661 pp;
Hardcover

Print ISBN: 978-1-93951-205-5

Product Code: TEXT/31

List Price: $75.00

AMS Member Price: $56.25

MAA Member Price: $56.25

**Electronic ISBN: 978-1-61444-617-0
Product Code: TEXT/31.E**

List Price: $75.00

AMS Member Price: $56.25

MAA Member Price: $56.25

# An Invitation to Real Analysis

Share this page
*Luis F. Moreno*

MAA Press: An Imprint of the American Mathematical Society

An Invitation to Real Analysis is
written both as a stepping stone to higher calculus and analysis
courses, and as foundation for deeper reasoning in applied
mathematics. This book also provides a broader foundation in real
analysis than is typical for future teachers of secondary
mathematics. In connection with this, within the chapters, students
are pointed to numerous articles from The College Mathematics
Journal and The American Mathematical Monthly. These
articles are inviting in their level of exposition and their
wide-ranging content. Axioms are presented with an emphasis on the
distinguishing characteristics that new ones bring, culminating with
the axioms that define the reals.

Set theory is another theme found in
this book, beginning with what students are familiar with from basic
calculus. This theme runs underneath the rigorous development of
functions, sequences, and series, and then ends with a chapter on
transfinite cardinal numbers and with chapters on basic point-set
topology. Differentiation and integration are developed with the
standard level of rigor, but always with the goal of forming a firm
foundation for the student who desires to pursue deeper study.

A historical theme interweaves throughout the book, with many
quotes and accounts of interest to all readers. Over 600 exercises and
dozens of figures help the learning process. Several topics (continued
fractions, for example), are included in the appendices as enrichment
material. An annotated bibliography is included.

An instructor's manual for this title is available
electronically. Please send email to textbooks@ams.org for more
information.

#### Reviews & Endorsements

The title of this book suggests a friendly tone and a gentle introduction to real analysis. This does indeed seem to be the case, as the book's size and reader-friendly layout suggest. … The annotated bibliography will be appreciated by both the instructor and by interested students.

-- CMS Notices

# Table of Contents

## An Invitation to Real Analysis

- Cover cov11
- Half title i3
- Copyright ii4
- Title iii5
- Epigraph iv6
- Series v7
- Dedication vii9
- Contents ix11
- To the Student xiii15
- To the Instructor xvii19
- 0 Paradoxes? 121
- 1 Logical Foundations 525
- 2 Proof, and the Natural Numbers 1737
- 3 The Integers, and the Ordered Field of Rational Numbers 2747
- 4 Induction and Well-Ordering 3757
- 5 Sets 4565
- 6 Functions 5777
- 7 Inverse Functions 7191
- 8 Some Subsets of the Real Numbers 7999
- 9 The Rational Numbers Are Denumerable 87107
- 10 The Uncountability of the Real Numbers 93113
- 11 The Infinite 97117
- 12 The Complete, Ordered Field of Real Numbers 111131
- 13 Further Properties of Real Numbers 121141
- 14 Cluster Points and Related Concepts 125145
- 15 The Triangle Inequality 131151
- 16 Infinite Sequences 135155
- 17 Limits of Sequences 141161
- 18 Divergence: The Non-Existence of a Limit 149169
- 19 Four Great Theorems in Real Analysis 157177
- 20 Limit Theorems for Sequences 167187
- 21 Cauchy Sequences and the Cauchy Convergence Criterion 175195
- 22 The Limit Superior and Limit Inferior of a Sequence 181201
- 23 Limits of Functions 187207
- 24 Continuity and Discontinuity 201221
- 25 The Sequential Criterion for Continuity 213233
- 26 Theorems About Continuous Functions 219239
- 27 Uniform Continuity 227247
- 28 Infinite Series of Constants 237257
- 29 Series with Positive Terms 251271
- 30 Further Tests for Series with Positive Terms 263283
- 31 Series with Negative Terms 273293
- 32 Rearrangements of Series 283303
- 33 Products of Series 291311
- 34 The Numbers e and γ 303323
- 35 The Functions exp x and ln x 313333
- 36 The Derivative 319339
- 37 Theorems for Derivatives 331351
- 38 Other Derivatives 341361
- 39 The Mean Value Theorem 351371
- 40 Taylor's Theorem 359379
- 41 Infinite Sequences of Functions 367387
- 42 Infinite Series of Functions 377397
- 43 Power Series 389409
- 44 Operations with Power Series 399419
- 45 Taylor Series 415435
- 46 Taylor Series, Part II 423443
- 47 The Riemann Integral 433453
- 48 The Riemann Integral, Part II 449469
- 49 The Fundamental Theorem of Integral Calculus 461481
- 50 Improper Integrals 475495
- 51 The Cauchy-Schwarz and Minkowski Inequalities 485505
- 52 Metric Spaces 489509
- 53 Functions and Limits in Metric Spaces 501521
- 54 Some Topology of the Real Number Line 509529
- 55 The Cantor Ternary Set 517537
- Appendix A Farey Sequences 527547
- Appendix B Proving that Σ(sup[n])(sub[k=0]) i/k! < (I + 1/n)(sup[n+1]) 531551
- Appendix C The Ruler Function Is Riemann Integrable 535555
- Appendix D Continued Fractions 539559
- Appendix E L'Hospital's Rule 545565
- Appendix F Symbols, and the Greek Alphabet 555575
- Annotated Bibliography 557577
- Solutions to Odd-Numbered Exercises 561581
- Index 655675