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### Introduction to Real Analysis, 3E, Robert G. Bartle, Donald R. Sherbert

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Introduction to Real Analysis, 3E written by Robert G. Bartle, Donald R. Sherbert
The first two editions of this book were very well received, and we have taken pains to maintain the same spirit and user-friendly approach. In preparing this edition, we have examined every section and set of exercises, streamlined some arguments, provided a few new examples, moved certain topics to new locations, and made revisions. Except for the new Chapter 10, which deals with the generalized Riemann integral, we have not added much new material. While there is more material than can be covered in one semester, instructors may wish to use certain topics as honors projects or extra credit assignments.
It is desirable that the student have had some exposure to proofs, but we do not assume that to be the case. To provide some help for students in analyzing proofs of theorems, we include an appendix on "Logic and Proofs" that discusses topics such as implications, quantifiers, negations, contrapositives, and different types of proofs. We have kept the discussion informal to avoid becoming mired in the technical details of formal logic. We feel that 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.
(Robert G. Bartle)

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, 3E 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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