Elements of Real Analysis is a monograph aimed at the senior undergraduate and first year graduate level introduction to real analysis. In 428 pages the authors expose the reader to topics from an introduction to real numbers and sequences and series to Lebesgue measure and integration. The material is divided into eleven chapters. The writing is informative without being too compelling.
The book will serve its intended purpose and audience well. It is accessible to advanced undergraduate students. The authors include enough exercises in each section to improve understanding of the material, though I think that some exercises could be omitted for more advanced problems. Unfortunately, some of the problems contain important definitions and concepts that should have been part of the main text. In some parts of the text there is a decided lack of examples for concepts like integration by parts. Some parts of the book can be challenging to students as the text contains minor mistakes, mainly concerning the interchangeable use of < and ≤.
In my opinion, the book is more of a theory of calculus than an introduction to real analysis because important concepts from the latter are omitted. The book lacks the construction of the real numbers, which are taken as given, and also introduces the idea of Lebesgue measure and integration only in the two last chapters. The book also lacks some of the historical comments that help to place the study of modern analysis in the context of calculus and mathematics in general. The book is recommended as a textbook for undergraduate courses in introduction to real analysis but the professor may have to supplement in some places.
Kai Brunkalla teaches at Walsh University in North Canton, Ohio.
Natural Numbers, Integers, Rational Numbers
Decimal Representation of Real Numbers
Sequences and Convergence
Properties of Convergent Sequences
The Cauchy Criterion
Upper and Lower Limits
Open and Closed Sets
LIMIT OF A FUNCTION
Limit of a Function
Some Extensions of the Limit
Combinations of Continuous Functions
Continuity on an Interval
Compact Sets and Continuity
TheMean Value Theorem
THE RIEMANN INTEGRAL
Darboux's Theorem and Riemann Sums
Properties of the Integral
The Fundamental Theorem of Calculus
SEQUENCES AND SERIES OF FUNCTIONS
Sequences of Functions
Properties of Uniform Convergence
Series of Functions
Classes of Subsets of R
Lebesgue Outer Measure
Definition of the Lebesgue Integral
Properties of the Lebesgue Integral
Lebesgue Integral and Pointwise Convergence
Lebesgue and Riemann Integrals