## Elements of Real Analysis, 1/e

Herbert S. Gaskill, Memorial University of Newfoundland
Pallasena P. Narayanaswami, Memorial University of Newfoundland

Published August, 1997 by Prentice Hall Engineering/Science/Mathematics

Cloth
ISBN 0-13-897067-X

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Real Analysis-Mathematics

Unique in approach, this text is specifically designed to help today's weaker students achieve competence in serious mathematics. This text develops mathematical sophistication with a thorough knowledge and understanding of limiting processes. Unlike other texts — which merely present the theory and follow with a set of exercises in a routine fashion — this text provides insight into the techniques of analysis — explaining at the end of each key definition, example, and theorem what students must do to come to grips with the concepts. Concepts are presented slowly and include the details of calculations as well as substantial explanations as to how and why one proceeds in the given manner.  Comprehensive in coverage, the text explores the principles of logic, the axioms for the real numbers, limits of sequences, limits of functions, differentiation and integration, infinite series, convergence, uniform convergence for sequences of real-valued functions — and treats several topics often slighted or omitted in other texts.
Emphasizes the understanding of fundamental and basic concepts before tackling unwieldy generalizations.
Addresses students' lack of skill at symbol manipulation and basic algebra head-on.

• Consistently stresses the role of algebra and algebraic manipulation in the construction of proofs.
• Follows each theorem with a standard type of mathematical proof — with calculations performed in great detail so students are not left with a feeling that mathematics is mysterious or that understanding is beyond their abilities.
• As the subject unfolds, the proofs may have “gaps” to be filled by the student.
Follows each definition, proof and example with a Discussion section which:
• Explains the essential piece of insight which is being presented.
• Presents material on intuition, on what thought processes might lead to the proof which was just presented, etc.
Uses the words WHY? and HOW? throughout — inviting students to become active participants and to supply a missing argument or a simple calculation.
Encourages students to think like mathematicians.
• Helps students view real analysis as a cohesive whole which provides answers to a series of generic questions.
• E.g., Each time a new limiting process is introduced, the question is asked of how the process behaves with respect to the algebraic operations on the underlying structure; plausible conjectures based on previous knowledge are generated; and then students are encouraged to review the previous theory and attempt to generate the questions and the answers.
Contains more than 1000 individual exercises.
• Some are very simple, requiring little more than the completion of the details of an argument or a simple calculation.
• Others are much more difficult in that they ask students to “sort out” a situation: to act like mathematicians and find the desired results or to structure an investigation to achieve the required end (e.g., to find a characterization of those situations in which a function is lower semicontinuous at a point at which it has a jump discontinuity).

Emphasizes the understanding of fundamental and basic concepts before tackling unwieldy generalizations.

Addresses students' lack of skill at symbol manipulation and basic algebra head-on.
• Consistently stresses the role of algebra and algebraic manipulation in the construction of proofs.
• Follows each theorem with a standard type of mathematical proof — with calculations performed in great detail so students are not left with a feeling that mathematics is mysterious or that understanding is beyond their abilities.
• As the subject unfolds, the proofs may have “gaps” to be filled by the student.
Follows each definition, proof and example with a Discussion section which:
• Explains the essential piece of insight which is being presented.
• Presents material on intuition, on what thought processes might lead to the proof which was just presented, etc.
Uses the words WHY? and HOW? throughout — inviting students to become active participants and to supply a missing argument or a simple calculation.
Encourages students to think like mathematicians.
• Helps students view real analysis as a cohesive whole which provides answers to a series of generic questions.
• E.g., Each time a new limiting process is introduced, the question is asked of how the process behaves with respect to the algebraic operations on the underlying structure; plausible conjectures based on previous knowledge are generated; and then students are encouraged to review the previous theory and attempt to generate the questions and the answers.
Contains more than 1000 individual exercises.
• Some are very simple, requiring little more than the completion of the details of an argument or a simple calculation.
• Others are much more difficult in that they ask students to “sort out” a situation: to act like mathematicians and find the desired results or to structure an investigation to achieve the required end (e.g., to find a characterization of those situations in which a function is lower semicontinuous at a point at which it has a jump discontinuity).
Stresses and reviews elementary algebra and symbol manipulation as essential tools for success at the kind of computations required in dealing with limiting processes.
• Emphasizes that algebraic calculations constitute the bulk of most proofs in elementary analysis and in advanced analysis as well — and thus eases students' difficulties at an early stage.
• Directly presents, or includes as exercises, the most important algebraic tools — e.g., proves the binomial theorem to illustrate proof by induction. Similarly treats basic methods of factoring, sum of a geometric series, methods for dealing with inequalities, etc.
Begins with an axiomatic development of the reals for a deep understanding of the real numbers.
• Develops the positive integers within the context of the axioms for a complete ordered field — an axiomatic foundation from which all proofs can be generated with no hand-waving.
• Includes an appendix on set theory.
Stresses and develops the delta-epsilon (…d — …e) definition as the basis of real analysis.
• Follows each presentation of a delta-epsilon definition with numerous examples and a series of problems which force students to apply the definition directly to specific cases — e.g., students are expected to be able to prove from first principles that f(x) = x^4 cannot be uniformly continuous on R but is uniformly continuous on [-2, 96].
• Provides a framework in which to perform the computations and emphasizes the relationship of the present computations to those which students have already mastered.
• Shows that in most cases the computations which are required amount to no more than high school algebra.
Emphasizes — through all stages and in many examples — the need for the correct formulation of a negation of key statements.
Includes a number of special topics and/or increased depth of treatment of selected topics often slighted or omitted in other texts, e.g.:
• Improper integrals, Riemann-Stieltjes integration and the Weierstrass Approximation theorem.
• Detailed exposition relative to the negation of the main definitions, delicate tests for convergence of series and the approach taken to the transcendental functions.

0. Basic Concepts.
1. Limits of Sequences.
2. Limits of Functions.
3. A Little Topology.
4. Differentiation.
5. Integration.
6. Infinite Series of Constants.
7. Sequences of Functions.
8. Infinite Series of Functions.
9. Transcendental Functions.
Appendix on Set Theory.