Real Analysis and Infinity

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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
6.4 Limits via neighborhoods: The ε and δ
6.5 The mean value theorem
6.6 Indeterminate forms and l'Hospital's rule
6.6.1 L'Hospital's rule: The 0/0 form
6.6.2 L'Hospital's rule: The ∞/∞ form
6.7 The Newton–Raphson approximation method
6.7.1 The Newton–Raphson recursion
6.7.2 Estimating roots of real numbers
6.7.3 Choosing the initial value
6.8 Exercises
7 Integration
7.1 From velocity to position, from curve to area
7.2 Riemann integration and integrable functions
7.2.1 Step 1: Interval partitions
7.2.2 Step 2: Riemann sums
7.2.3 Step 3: Taking limits (invoking infinity)
7.2.4 Integrability of continuous functions and monotone functions
7.2.5 A non-Riemann integrable function
7.3 Properties of the integral
7.4 The Fundamental Theorem of Calculus
7.5 Logarithmic and exponential functions
7.5.1 The natural logarithm function
7.5.2 The natural exponential function
7.5.3 The general exponential and logarithmic functions
7.6 The improper Riemann integral
7.6.1 Unbounded functions
7.6.2 Unbounded intervals
7.7 The integral test for infinite series
7.8 Exercises
8 Infinite Sequences and Series of Functions
8.1 The geometric series as a function series
8.2 Convergence of an infinite sequence of functions
8.2.1 Pointwise convergence
8.2.2 Uniform convergence
8.2.3 On preserving limits, integrals, and derivatives
8.2.4 Uniform convergence does not preserve lengths
8.3 Infinite series of functions
8.3.1 The Cauchy criterion and the Weierstrass test
8.3.2 Integration and differentiation term by term of function series
8.3.3 Weierstrass's continuous yet nowhere differentiable function
8.4 Power series expansions of functions
8.4.1 Power series and their convergence
8.4.2 Continuity and integrability
8.4.3 Taking the derivative of a power series
8.5 The Taylor series expansion of functions
8.5.1 Infinitely differentiable functions and power series
8.5.2 Taylor polynomials and approximation
8.5.3 The remainder, approximation error, and convergence of Taylor polynomials
8.5.4 The binomial series and its applications
8.6 Trigonometric series and Fourier expansions: A glance
8.6.1 Trigonometric series
8.6.2 Fourier series and their coefficients
8.7 The Weierstrass approximation theorem
8.8 Exercises
9 Appendices
9.1 Appendix: Cantor's construction, additional detail
9.2 Appendix: Discontinuity in a space of functions
9.3 Numbered Elements in every chapter
References and Further Reading
Index