Introduction to Real Analysis: An Educational Approach

Acknowledgments.

1 Elementary Calculus.

1.1 Preliminary Concepts.

1.2 Limits and Continuity.

1.3 Differentiation.

1.4 Integration.

1.5 Sequences and Series of Constants.

1.6 Power Series and Taylor Series.

Summary.

Exercises.

Interlude: Fermat, Descartes, and theTangent Problem.

2 Introduction to Real Analysis.

2.1 Basic Topology of the Real Numbers.

2.2 Limits and Continuity.

2.3 Differentiation.

2.4 Riemann and Riemann-Stieltjes Integration.

2.5 Sequences, Series, and Convergence Tests.

2.6 Pointwise and Uniform Convergence.

Summary.

Exercises.

Interlude: Euler and the "Basel Problem".

3 A Brief Introduction to Lebesgue Theory.

3.1 Lebesgue Measure and Measurable Sets.

3.2 The Lebesgue Integral.

3.3 Measure, Integral, and Convergence.

3.4 Littlewood’s Three Principles.

Summary.

Exercises.

Interlude: The Set of Rational Numbers isVery Large andVery Small.

4 Special Topics.

4.1 Modeling with Logistic Functions—Numerical Derivatives.

4.2 Numerical Quadrature.

4.3 Fourier Series.

4.4 Special Functions—The Gamma Function.

4.5 Calculus Without Limits: Differential Algebra.

Summary.

Exercises.

Appendix A: Definitions and Theorems of Elementary Real Analysis.

A.1 Limits.

A.2 Continuity.

A.3 The Derivative.

A.4 Riemann Integration.

A.5 Riemann-Stieltjes Integration.

A.6 Sequences and Series of Constants.

A.7 Sequences and Series of Functions.

Appendix B: A Very Brief Calculus Chronology.

Appendix C: Projects in Real Analysis.

C.1 Historical Writing Projects.

C.2 Induction Proofs: Summations, Inequalities, and Divisibility.

C.3 Series Rearrangements.

C.4 Newton and the Binomial Theorem.

C.5 Symmetric Sums of Logarithms.

C.6 Logical Equivalence: Completeness of the Real Numbers.

C.7 Vitali’s Nonmeasurable Set.

C.8 Sources for Real Analysis Projects.

C.9 Sources for Projects for Calculus Students.

Bibliography.

Index.