Preface
Some basic notation

Chapter 1. Numbers
1.1. Peano arithmetic
1.2. The integers
1.3. Prime factorization and the fundamental theorem of arithmetic
1.4. The rational numbers
1.5. Sequences
1.6. The real numbers
1.7. Irrational numbers
1.8. Cardinal numbers
1.9. Metric properties of RR
1.10. Complex numbers

Chapter 2. Spaces
2.1. Euclidean spaces
2.2. Metric spaces
2.3. Compactness
2.4. The Baire category theorem

Chapter 3. Functions
3.1. Continuous functions
3.2. Sequences and series of functions
3.3. Power series
3.4. Spaces of functions
3.5. Absolutely convergent series

Chapter 4. Calculus
4.1. The derivative
4.2. The integral
4.3. Power series
4.4. Curves and arc length
4.5. The exponential and trigonometric functions
4.6. Unbounded integrable functions

Chapter 5. Further topics in analysis
5.1. Convolutions and bump functions
5.2. The Weierstrass approximation theorem
5.3. The Stone–Weierstrass theorem
5.4. Fourier series
5.5. Newton’s method
5.6. Inner product spaces

Appendix A. Complementary results
A.1. The fundamental theorem of algebra
A.2. More on the power series of (1-x)^{b}
A.3. π² is irrational
A.4. Archimedes’ approximation to π
A.5. Computing π using arctangents
A.6. Power series for tanx
A.7. Abel’s power series theorem
A.8. Continuous but nowhere-differentiable functions

Bibliography
Index