Bartle and Sherbert's classic Introduction To Real Analysis gives a rigorous development of real analysis in one variable. Analysis is a branch of mathematics that justifies and proves all the techniques and results of differential & integral calculus. It deals with concepts such as smoothness, convergence, divergence, and so on.
Their treatment of limits, of continuity, of convergence, of differentiation and integration is exact and complete. They give readers a full grounding in epsilon/delta proof methodology for the major theorems of modern single variable calculus.
Because they deal in a single variable, they don't spend much time on basic topology. The book consists of eight chapters. A brief introduction to set theory is followed by a presentation of the real number system. Note that they don't construct the field of real numbers, they merely state the completeness theorem that fills in the gaps found in the field of rational numbers (e.g. the square root of two is a real number not found in the rationals).
The meat of the book begins with chapter three on sequences followed by chapters on limits & continuity, differentiation, Riemann integration, sequences of functions, and finally infinite series.
The many exercises will give readers much opportunity to hone their skills.
I have a few pet peeves. I find the tone a little patronizing. Walter Rudin's Principles of Mathematical Analysis is much more rigorous and explores the topic in greater depth than does Bartle & Sherbert's textbook, but he nowhere adopts their slightly consdescending tone.
Also, the presentation is a little dry. Many of the theorems they give are profound and exciting but one doesn't get this from the text. And they miss out on even hinting at fascinating results because it falls outside the scope of their program. For example, they spend a great deal of time on a rigorous elaboration the sine as cosine functions purely through their derivative properties, with no reference to their geometry interpretation. But because their text doesn't deal with complex numbers, they miss out on presenting a beautiful result that follows straightforwardly from this construction.
Overall, a solid and correct but not very inspiring introduction to the topic. Still, this is a great book from which to teach a course. Teachers can supply the inspiration themselves.
Vincent Poirier, Tokyo