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Description: Elementary Real Analysis is a core course in nearly all college and university mathematics departments. It enables students to develop a deep understanding of the key concepts of calculus from a mature perspective.
Elements of Real Analysis is a student-friendly guide to learning all the important ideas of elementary real analysis, based on the author’s many years of experience teaching the subject to typical undergraduates. It avoids the compact style of professional mathematics writing, in favor of a style that feels more comfortable to students encountering the subject for the first time. It presents topics in ways that are most easily understood, without sacrificing rigor or coverage. Important concepts are discussed in greater detail than customary in a text at this level.
Using this text, students discover that real analysis is completely deducible from the axioms of the real number system. They learn the powerful techniques of limits of sequences as the primary entry to the concepts of analysis, and see the ubiquitous role sequences play in virtually all later topics. They become comfortable with topological ideas, and see how these concepts help unify the subject. They develop a unified understanding of limits, continuity, differentiability, Riemann integrability, and infinite series of numbers and functions. Students encounter many interesting examples, including “pathological” ones, that motivate the subject and help fix the concepts.Key features:
• Flexible in design, the text can be used in a one-or two-term course
• A student friendly conversational writing style helps students grasp difficult concepts
• Proofs are written in a style appropriate for undergraduates to emulate in their homework
• Numerous examples throughout the book clearly illustrate the concepts at hand.
• Generous exercise sets, including many routine exercises, help students develop confidence
• Many project-type exercises guide students in exploring advanced topics
• An appendix includes answers and hints for selected exercisesContents:
Chapter 1. The Real Number System • The Field Properties • The Order Properties • Natural Numbers • Rational Numbers • The Archimedean Property • The Completeness Property • “The” Complete Ordered Field
Chapter 2. Sequences • Basic Concepts: Convergence and Limits • Algebra of Limits • Inequalities and Limits • Divergence to Infinity • Monotone Sequences • Subsequences and Cluster Points • Cauchy Sequences • Countable and Uncountable Sets • Upper and Lower Limits
Chapter 3. Topology of the Real Number System • Neighborhoods and Open Sets • Closed Sets and Cluster Points • Compact Sets • The Cantor Set
CHapter 4. Limits of Functions • Definition of Limit for Functions • Algebra of Limits of Functions • One-Sided Limits • Infinity in Limits
Chapter 5. Continuous Functions • Continuity of a Function at a Point • Discontinuities and Monotone Functions • Continuity on Compact Sets and Intervals • Uniform Continuity . • Monotonicity, Continuity, and Inverses • Exponentials, Powers, and Logarithms • Sets of Points of Discontinuity (Project)
Chapter 6. Differentiable Functions • The Derivative and Differentiability • Rules for Differentiation • Local Extrema and Monotone Functions • Mean-Value Type Theorems • Taylor’s Theorem • L’Hˆopital’s Rule
Chapter 7. The Riemann Integral • Refresher on Suprema, Infima, and the Forcing Principle • The Riemann Integral Defined • The Integral as a Limit of Riemann Sums • Basic Existence and Additivity Theorems • Algebraic Properties of the Integral • The Fundamental Theorem of Calculus • Elementary Transcendental Functions • Improper Riemann Integrals • Lebesgue’s Criterion for Riemann Integrability
Chapter 8. Infinite Series of Real Numbers • Basic Concepts and Examples . • Nonnegative Series • Series with Positive and Negative Terms • The Cauchy Product of Series • Series of Products • Power Series • Analytic Functions • Elementary Transcendental Functions (Project)
Chapter 9. Sequences and Series of Functions • Families of Functions and Pointwise Convergence • Uniform Convergence • Implications of Uniform Convergence in Calculus • Two Results of Weierstrass • A Glimpse Beyond the Horizon
APPENDICES • A Logic and Proofs • A. The Logic of Propositions • A. The Logic of Predicates and Quantifiers • A. Strategies of Proving Theorems • A. Properties of Equality • B Sets and Functions • B. Sets and the Algebra of Sets • B. Functions • B. Algebra of Real-Valued Functions • C Answers & Hints for Selected Exercises • Bibliography • Glossary of Symbols • IndexISBN - 9789380853154
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Pages : 468
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