Introduction to Real Analysis

Citation

Bartle, R. G., & Sherbert, D. R. (2011). Introduction to Real Analysis (4th ed.). John Wiley & Sons.

Chapter Summary

Chapter 1: Preliminaries

Three sections supply the working vocabulary: sets and functions, mathematical induction, and finite and infinite sets. Appendix A on logic and proofs and Appendix B on finite and countable sets back this material up.

Chapter 2: The Real Numbers

The chapter builds the real line from its properties: the algebraic and order properties, absolute value and the real line, and the completeness property. A section on applications of the supremum property and one on intervals close the chapter.

Chapter 3: Sequences and Series

Sequences and their limits, limit theorems, monotone sequences, subsequences and the Bolzano–Weierstrass theorem, the Cauchy criterion and properly divergent sequences make up the first six sections. A final section gives an introduction to infinite series, placing series directly after sequences.

Chapter 4: Limits

Limits of functions, the limit theorems for them, and some extensions of the limit concept prepare for continuity.

Chapter 5: Continuous Functions

Continuity and combinations of continuous functions lead to continuous functions on intervals, uniform continuity and, unusually, continuity and gauges. The chapter closes with monotone and inverse functions.

Chapter 6: Differentiation

The derivative, the mean value theorem, L'Hospital's rules and Taylor's theorem.

Chapter 7: The Riemann Integral

The Riemann integral, Riemann integrable functions, the fundamental theorem, the Darboux integral and approximate integration.

Chapter 8: Sequences of Functions

Pointwise and uniform convergence and the interchange of limits come first. The exponential and logarithmic functions, and then the trigonometric functions, are developed in this setting.

Chapter 9: Infinite Series

Absolute convergence, tests for absolute convergence and for nonabsolute convergence, and series of functions.

Chapter 10: The Generalized Riemann Integral

The gauge (generalized Riemann) integral: its definition and main properties, the relation to improper and Lebesgue integrals, infinite intervals and convergence theorems.

Chapter 11: A Glimpse into Topology

Open and closed sets in R, compact sets, continuous functions in this setting, and metric spaces.

Appendices

Appendix A covers logic and proofs, B finite and countable sets, C the Riemann and Lebesgue criteria, D approximate integration, and E two examples. Hints for selected exercises occupy the last twenty-odd pages before the index.

Key Concepts

Chapters 1–2: Preliminaries and the Real Numbers

• Mathematical induction: the proof method for statements about natural numbers.
• Completeness (supremum) property: the property that distinguishes the real numbers from the rationals.

Chapters 3–5: Sequences, Limits and Continuity

• Bolzano–Weierstrass theorem and the Cauchy criterion: two ways to conclude that a sequence has a limit or convergent subsequence.
• Uniform continuity: continuity with a single tolerance valid on a whole set.
• Gauges: the device that links continuity in Chapter 5 to the generalized integral in Chapter 10.

Chapters 6–7: Differentiation and the Riemann Integral

• Mean value theorem and Taylor's theorem: the main consequences of differentiability.
• Fundamental theorem of calculus and the Darboux integral: the link between differentiation and integration, and an alternative definition of the integral.

Chapters 8–11: Convergence, Generalized Integration and Topology

• Uniform convergence: the condition that lets limits be interchanged with continuity, differentiation and integration.
• Generalized Riemann integral: a more powerful integral that contains the Riemann and Lebesgue integrals.
• Compact sets and metric spaces: the topological ideas that take the theory beyond the real line.

Critical Analysis

Strengths

• A stated teaching approach. Wiley describes the book as helping students think deductively, analyze mathematical situations and extend ideas to a new context, in the spirit of the earlier editions.
• Foundations inside the book. The first chapter, Appendix A (logic and proofs) and Appendix B (finite and countable sets) mean that a student does not need a separate proof course before starting.
• A well-ordered single-variable course. The order runs from real numbers to sequences, limits, continuity, differentiation and integration, then sequences of functions and series. Series are introduced directly after sequences, a change made in this edition.
• A distinctive integration theory. The gauge viewpoint appears early, in the section on continuity and gauges, and returns in the full chapter on the generalized Riemann integral, which Wiley notes now includes the Lebesgue integral.
• A bridge to the next course. The final chapter introduces open, closed and compact sets and metric spaces in four sections.
• Self-study help. Hints for selected exercises run to over twenty pages at the back.

Limitations

• Single variable only. The contents stay on the real line, with metric spaces only glimpsed in the last chapter. Multivariable analysis, measure theory and complex analysis are not covered.
• Hints, not full solutions. The back matter offers hints for selected exercises, so independent learners cannot check complete arguments.
• Specialised late chapters. The generalized Riemann integral (Chapter 10) and topology (Chapter 11) come after nine chapters of core material. Readers who need a standard measure-theoretic treatment of the Lebesgue integral will still need another book.

Where It Fits in the Course

Bartle and Sherbert is the comprehensive conventional textbook for a first rigorous analysis course in the real analysis list. Alcock's How to Think About Analysis is a study guide that explains how to read definitions and build proofs, and is a good companion for a student starting this book. Burn's Numbers and Functions: Steps into Analysis teaches almost entirely through carefully sequenced problems. Abbott's Understanding Analysis is a textbook for a first rigorous course in which each chapter opens with a motivating discussion and closes with a project. Bryant's Yet Another Introduction to Analysis is a further introduction on the list.

For a more demanding treatment, Rudin's Principles of Mathematical Analysis is terse and covers metric spaces, functions of several variables and the Lebesgue integral. Bartle and Sherbert is a gentler path into the same single-variable material and a natural preparation for Rudin.

How to Use This Book

The book is for mathematics, science and engineering students taking their first proof-based course in analysis, who should have completed single-variable calculus.

A one-semester course normally takes Chapters 1 to 7: use Chapter 1 and the appendices as needed, then work through Chapters 2 to 7 in order. A second semester continues with Chapters 8 and 9, and then Chapter 10 or Chapter 11 according to taste. Chapter 11 is a good bridge for students heading to topology or metric-space analysis; Chapter 10 is for those interested in integration theory.

Do the exercises, especially the early ones in each section, and compare with the hints only after an honest attempt. Reading Alcock alongside Chapters 2 to 5 is helpful for students who find the transition to proofs difficult.