
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
Copyright 1998, 501 pp.
Cloth
ISBN 013897067X

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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 realvalued
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 headon.
 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 headon.
 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 handwaving.
 Includes an appendix on set theory.
Stresses and develops the deltaepsilon (…d —
…e) definition as the basis of real analysis.
 Follows each presentation of a deltaepsilon 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, RiemannStieltjes 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.
Suggested Reading.
Index.
