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### Introduction to Real Analysis, 4E, Robert G. Bartle, Donald R. Sherbert

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Introduction to Real Analysis, 4E written by Robert G. Bartle, Donald R. Sherbert
The first three editions were very well received and this edition maintains the same spirit and user-friendly approach as earlier editions. Every section has been examined.
Some sections have been revised, new examples and exercises have been added, and a new section on the Darboux approach to the integral has been added to Chapter 7. There is more material than can be covered in a semester and instructors will need to make selections and perhaps use certain topics as honors or extra credit proj ects.
To provide some help for students in analyzing proofs of theorems, there is an appendix on "Logic and Proofs" that discusses topics such as implications, negations, contrapositives, and different types of proofs. However, it is a more useful experience to learn how to construct proofs by first watching and then doing than by reading about techniques of proof.
Results and proofs are given at a medium level of generality. For instance, continuous functions on closed, bounded intervals are studied in detail, but the·proofs can be readily adapted to a more general situation. This approach is used to advantage in Chapter 11 where topological concepts are discussed. There are a large number of examples to illustrate the concepts, and extensive lists of exercises to challenge students and to aid them in understanding the significance of the theorems.
(Robert G. Bartle)

Book Detail :-
Title: Introduction to Real Analysis
Edition: 4th
Author(s): Robert G. Bartle, Donald R. Sherbert
Publisher: John Wiley & Sons
Series:
Year: 2011
Pages: 417
Type: PDF
Language: English
ISBN: 0471433314,9780471433316
Country: US

Author Robert Gardner Bartle (1927–2003) was an American mathematician. He was specializing in real analysis. He is known for writing the popular textbooks The Elements of Real Analysis (1964), The Elements of Integration (1966), and Introduction to Real Analysis (2011) published by John Wiley & Sons.
Bartle was born in Kansas City, Missouri, and was the son of Glenn G. Bartle and Wanda M. Bartle. He was married to Doris Sponenberg Bartle (born 1927) from 1952 to 1982 and they had two sons, James A. Bartle (born 1955) and John R. Bartle (born 1958). He was on the faculty of the Department of Mathematics at the University of Illinois from 1955 to 1990.
Bartle was Executive Editor of Mathematical Reviews from 1976 to 1978 and from 1986 to 1990. From 1990 to 1999 he taught at Eastern Michigan University. In 1997, he earned a writing award from the Mathematical Association of America for his paper "Return to the Riemann Integral".[1]

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Book Contents :-
Introduction to Real Analysis, 4E written by Robert G. Bartle, Donald R. Sherbert cover the following topics.
1. PRELIMINARIES
Sets and Functions, Mathematical Induction, Finite and Infinite Sets
2. THE REAL NUMBERS
The Algebraic and Order Properties of R, Absolute Value and the Real Line, The Completeness Property of R, Applications of the Supremum Property, Intervals
3. SEQUENCES AND SERIES
Sequences and Their Limits, Limit Theorems, Monotone Sequences, Subsequences and the Bolzano-Weierstrass Theorem, The Cauchy Criterion, Properly Divergent Sequences, Introduction to Infinite Series
4. LIMITS
Limits of Functions, Limit Theorems, Some Extensions of the Limit Concept
5. CONTINUOUS FUNCTIONS
Continuous Functions, Combinations of Continuous Functions, Continuous Functions on Intervals, Uniform Continuity, Continuity and Gauges, Monotone and Inverse Functions
6. DIFFERENTIATION
The Derivative, The Mean Value Theorem, L’Hospital’s Rules, Taylor’s Theorem
7. THE RIEMANN INTEGRAL
Riemann Integral, Riemann Integrable Functions, The Fundamental Theorem, The Darboux Integral, Approximate Integration
8. SEQUENCES OF FUNCTIONS
Pointwise and Uniform Convergence, Interchange of Limits, The Exponential and Logarithmic Functions, The Trigonometric Functions
9. INFINITE SERIES
Absolute Convergence, Tests for Absolute Convergence, Tests for Nonabsolute Convergence, Series of Functions
10. THE GENERALIZED RIEMANN INTEGRAL
Definition and Main Properties, Improper and Lebesgue Integrals, Infinite Intervals, Convergence Theorems
11. A GLIMPSE INTO TOPOLOGY
Open and Closed Sets in R, Compact Sets, Continuous Functions, Metric Spaces
APPENDIX
LOGIC AND PROOFS, FINITE AND COUNTABLE SETS, THE RIEMANN AND LEBESGUE CRITERIA, APPROXIMATE INTEGRATION, TWO EXAMPLES

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