http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
Real Analysis and Infinity presents the essential topics for a first course in real analysis with an emphasis on the role of infinity in all of the fundamental concepts.
Cover -- titlepage -- copyright -- Contents -- Preface -- 1 Manifestations of Infinity: An Overview -- 1.1 Infinity within each number -- 1.2 More than one infinity? -- 1.3 Infinite processes -- 1.4 Trigonometric functions and logarithms: Infinity built in -- 1.5 Exercises -- 2 Sets, Functions, Logic, and Countability -- 2.1 Sets and relations -- 2.1.1 Set operations -- 2.1.2 Relations -- 2.2 Functions and their basic properties -- 2.2.1 Operations on functions -- 2.2.2 Image and inverse image sets -- 2.2.3 One-to-one functions and bijections -- 2.3 Basics of logic concepts and operations
2.3.1 Fundamental connectives and truth tables -- 2.3.2 Converse, contrapositive, and contradiction -- 2.3.3 Rules of inference -- 2.4 Mathematical induction -- 2.5 Bijections and cardinality -- 2.6 An infinity of infinities: Cantor's theorem -- 2.7 Countable or uncountable? -- 2.8 Exercises -- 3 Sequences and Limits -- 3.1 Infinite lists of numbers -- 3.2 Sequence types and plots -- 3.3 Monotone sequences and oscillating sequences -- 3.4 Convergent sequences and limits -- 3.5 Bounded, unbounded, and divergent sequences -- 3.6 Subsequences -- 3.7 Limit supremum and limit infimum
3.8 Sequences, functions, and infinite direct products of sets -- 3.9 Exercises -- 4 The Real Numbers -- 4.1 Rational numbers -- 4.2 Cauchy sequences -- 4.2.1 Equivalent Cauchy sequences -- 4.3 Real numbers -- 4.4 Completeness and other foundational theorems of real analysis -- 4.4.1 Completeness of R and the Cauchy convergence criterion -- 4.4.2 Density of rational numbers in R -- 4.4.3 Least upper bounds and nested intervals -- 4.4.4 The Bolzano-Weierstrass theorem -- 4.5 The set of real numbers is uncountable -- 4.6 Exercises -- 5 Infinite Series of Constants
5.1 On adding infinitely many numbers -- 5.2 Infinite series as limits of sequences of finite sums -- 5.3 The geometric series -- 5.4 Cauchy criterion and convergence tests -- 5.4.1 The Cauchy criterion -- 5.4.2 The comparison test -- 5.4.3 The ratio and root tests: Extending the geometric series method -- 5.5 Alternating series, conditional and absolute convergence -- 5.5.1 The alternating series -- 5.5.2 Absolute convergence -- 5.5.3 Conditional convergence and rearrangements of series -- 5.6 Real numbers as infinite series, Liouville numbers -- 5.7 Exercises -- 6 Differentiation and Continuity
6.1 Velocity, slope, and the derivative -- 6.1.1 Velocity and slope -- 6.1.2 The derivative -- 6.2 Differentiation rules and higher derivatives -- 6.2.1 Derivatives of sums, products, and quotients -- 6.2.2 The chain rule -- 6.2.3 Derivatives of trigonometric functions -- 6.2.4 Higher-order derivatives -- 6.2.5 When derivatives fail to exist -- 6.3 Continuous functions -- 6.3.1 Continuity and limits -- 6.3.2 Continuity and algebraic operations -- 6.3.3 The intermediate value theorem -- 6.3.4 Boundedness and the extreme value theorem -- 6.3.5 Uniform continuity
Includes bibliographical references and index.
Oxford, United Kingdom : Oxford University Press, [2022]