This book occupies a niche between a calculus course and a full-blown real analysis course. Its charm is that it gives very thorough and leisurely explanations, in a discursive style: You just read along about some interesting properties of the real numbers and then find them codified as a definition or theorem, rather than being confronted with a mass of definitions, theorems, and proofs.
The book assumes the student has already been through calculus, but without the proofs. It presents most of the important ideas of real analysis without requiring any great conceptual leaps from a calculus course. It is very carefully positioned to lie between a non-rigorous calculus course and a real analysis course such as might be taught from Rudin's Principles of Mathematical Analysis or Apostol's Mathematical Analysis.
The book is mathematically not very ambitious, and at first glance it may look like there's not much here. I think the book should be viewed as a text for a bridge or transition course that happens to be about analysis, rather than an analysis course per se.
Certainly it has much less material than would normally be found in an analysis text. There's not much topology, no construction of the real numbers (there's a brief sketch of Dedekind cuts), no measure theory or Lebesgue integral, and no function spaces. The book depends almost totally on completeness of the reals for its proofs, although the Bolzano-Weierstrass theorem is introduced to back up uniform continuity, which is needed for integrability.
There are several Very Good Features:
The book has a number of optional sections, which tend to be dead ends in this book but are interesting in themselves and in a more advanced course would have many consequences. Most of the topology material is in this category. There's a complete proof of the Weierstrass approximation theorem (using Bernstein polynomials), which doesn't go anywhere in this book but is certainly a startling result.
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.
1 The Set N of Natural Numbers
2 The Set Q of Rational Numbers
3 The Set R of Real Numbers
4 The Completeness Axiom
5 The Symbols +∞ and –∞
6 *A Development of R
7 Limits of Sequences
8 A Discussion about Proofs
9 Limit Theorems for Sequences
10 Monotone Sequences and Cauchy Sequences
12 lim sup's and lim inf's
13 *Some Topological Concepts in Metric Spaces
15 Alternating Series and Integral Tests
16 *Decimal Expansions of Real Numbers
17 Continuous Functions
18 Properties of Continuous Functions
19 Uniform Continuity
20 Limits of Functions
21 *More on Metric Spaces: Continuity
22 *More on Metric Spaces: Connectedness
Sequences and Series of Functions
23 Power Series
24 Uniform Convergence
25 More on Uniform Convergence
26 Differentiation and Integration of Power Series
27 *Weierstrass's Approximation Theorem
28 Basic Properties of the Derivative
29 The Mean Value Theorem
30 *L'Hospital's Rule
31 Taylor's Theorem
32 The Riemann Integral
33 Properties of the Riemann Integral
34 Fundamental Theorem of Calculus
35 *Riemann-Stieltjes Integrals
36 * Improper Integrals
37 *A Discussion of Exponents and Logarithms
Appendix on Set Notation
Selected Hints and Answers