Introduction to Real Analysis 4th Edition, by Robert G. Bartle and Donald R. Sherbert provides the fundamental concepts and techniques of real analysis for students in all of these areas. It helps one develop the ability to think deductively, analyze mathematical situations and extend ideas to a new context.
This text maintains the same spirit and user-friendly approach with addition examples and expansion on Logical Operations and Set Theory. There is also content revision in the following areas: introducing point-set topology before discussing continuity, including a more thorough discussion of limsup and limimf, covering series directly following sequences, adding coverage of Lebesgue Integral and the construction of the reals, and drawing student attention to possible applications wherever possible.
Several new examples have been added to this edition to make the text more up-to-date and relevant. New exercises have been added throughout to give students more material to practice and solidify their understanding of the material. Coverage of the Darboux integral has been added in Section 7.4. Analysis is a branch of mathematics that justifies and proves all the techniques and results of differential & integral calculus. It deals with concepts such as smoothness, convergence, divergence, and so on.
Their treatment of limits, of continuity, of convergence, of differentiation and integration is exact and complete. They give readers a full grounding in epsilon/delta proof methodology for the major theorems of modern single variable calculus.
Because they deal in a single variable, they don't spend much time on basic topology. The book consists of eight chapters. A brief introduction to set theory is followed by a presentation of the real number system. Note that they don't construct the field of real numbers, they merely state the completeness theorem that fills in the gaps found in the field of rational numbers (e.g. the square root of two is a real number not found in the rational).
The meat of the book begins with chapter three on sequences followed by chapters on limits & continuity, differentiation, Riemann integration, sequences of functions, and finally infinite series. The many exercises will give readers much opportunity to hone their skills.
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