An Invitation to Real Analysis (Mathematical Association of America Textbooks)

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. Investigation Of The Theorems And Methods That Establish Calculus. It Is A Textbook For An Analysis Course Following Calculus Ii. It Emphasizes The Writing Of Proofs, And Extends The Development Of Limits, Sequences, And Infinite Series. Riemann Integration Is Discussed In Depth. The Axiomatic Structures Presented Here Build Upward From Those For Natural Numbers To Those For Real Numbers. Related Topics Will Interest Those Who Need A Theoretical Background For Secondary School Teaching, The Sciences, And The Technologies. Historical Material Is Included Throughout The Text ... . Many Diagrams And Hints Support Students As They Navigate Through The Topics. Detailed Solutions To Odd-numbered Exercises Are Included.--back Cover. Paradoxes? -- Logical Foundations -- Proof, And The Natural Numbers -- The Integers, And The Ordered Field Of Rational Numbers -- Induction And Well-ordering -- Sets -- Functions -- Inverse Functions -- Some Subsets Of The Real Numbers -- The Rational Numbers Are Denumerable -- The Uncountability Of The Real Numbers -- The Infinite -- The Complete, Ordered Field Of Real Numbers -- Further Properties Of Real Numbers -- Cluster Points And Related Concepts -- The Triangle Inequality -- Infinite Sequences -- Limits Of Sequences -- Divergence : The Non-existence Of A Limit -- Four Great Theorems In Real Analysis -- Limit Theorems For Sequences -- Cauchy Sequences And The Cauchy Convergence Criterion -- The Limit Superior And Limit Inferior Of A Sequence -- Limits Of Functions -- Continuity And Discontinuity -- The Sequential Criterion For Continiuty -- Theorems About Continuous Functions -- Uniform Continuity -- Infinite Series Of Constants -- Series With Positive Terms -- Further Tests For Series With Positive Terms -- Series With Negative Terms -- Rearrangements Of Series -- Products Of Series -- The Numbers E And Y -- The Functions Exp X And In X -- The Derivative -- Theorems For Derivatives -- Other Derivatives -- The Mean Value Theorem -- Taylor's Theorem -- Infinite Sequences Of Functions -- Infinite Series Of Functions -- Power Series -- Operations With Power Series -- Taylor Series -- Taylor Series, Part Ii -- The Riemann Integral -- The Riemann Integral, Part Ii -- The Fundamental Theorem Of Integral Calculus -- Improper Integrals -- The Cauchy-schwarz And Minkowski Inequalities -- Metric Spaces -- Functions And Limits In Metric Spaces -- Some Topology Of The Real Number Line -- The Cantor Ternary Set. Luis F. Moreno. Includes Bibliographical References And Index. Cover Half title Copyright Title Epigraph Series Dedication Contents To the Student To the Instructor 0 Paradoxes? 1 Logical Foundations 2 Proof, and the Natural Numbers 3 The Integers, and the Ordered Field of Rational Numbers 4 Induction and Well-Ordering 5 Sets 6 Functions 7 Inverse Functions 8 Some Subsets of the Real Numbers 9 The Rational Numbers Are Denumerable 10 The Uncountability of the Real Numbers 11 The Infinite 12 The Complete, Ordered Field of Real Numbers 13 Further Properties of Real Numbers 14 Cluster Points and Related Concepts 15 The Triangle Inequality 16 Infinite Sequences 17 Limits of Sequences 18 Divergence: The Non-Existence of a Limit 19 Four Great Theorems in Real Analysis 20 Limit Theorems for Sequences 21 Cauchy Sequences and the Cauchy Convergence Criterion 22 The Limit Superior and Limit Inferior of a Sequence 23 Limits of Functions 24 Continuity and Discontinuity 25 The Sequential Criterion for Continuity 26 Theorems About Continuous Functions 27 Uniform Continuity 28 Infinite Series of Constants 29 Series with Positive Terms 30 Further Tests for Series with Positive Terms 31 Series with Negative Terms 32 Rearrangements of Series 33 Products of Series 34 The Numbers e and γ 35 The Functions exp x and ln x 36 The Derivative 37 Theorems for Derivatives 38 Other Derivatives 39 The Mean Value Theorem 40 Taylor's Theorem 41 Infinite Sequences of Functions 42 Infinite Series of Functions 43 Power Series 44 Operations with Power Series 45 Taylor Series 46 Taylor Series, Part II 47 The Riemann Integral 48 The Riemann Integral, Part II 49 The Fundamental Theorem of Integral Calculus 50 Improper Integrals 51 The Cauchy-Schwarz and Minkowski Inequalities 52 Metric Spaces 53 Functions and Limits in Metric Spaces 54 Some Topology of the Real Number Line 55 The Cantor Ternary Set Appendix A Farey Sequences Appendix B Proving that Σ(sup[n])(sub[k=0]) i/k! < (I + 1/n)(sup[n+1]) Appendix C The Ruler Function Is Riemann Integrable Appendix D Continued Fractions Appendix E L'Hospital's Rule Appendix F Symbols, and the Greek Alphabet Annotated Bibliography Solutions to Odd-Numbered Exercises Index Back cover This book is written as both a stepping stone to higher calculus and analysis courses, and as a foundation for deeper reasoning in applied mathematics. As well as a rigorous account of sequences, series, functions and sets, the reader will also find fascinating historical material and over 600 exercises.