This text evolved from notes developed for use in a two-semester undergraduate
course on foundations of analysis at the University of Utah. The course is designed
for students who have completed three semesters of calculus and one semester of
linear algebra. For most of them, this is the first mathematics course in which
everything is proved rigorously and they are expected to not only understand proofs
but to also create proofs.
The course has two main goals. The first is to develop in students the mathe-
matical maturity and sophistication they will need when they move on to senior or
graduate level mathematics courses. The second is to present a rigorous develop-
ment of the calculus, beginning with a study of the properties of the real number
We have tried to present this material in a fashion which is both rigorous and
concise, with simple, straightforward explanations. We feel that the modern ten-
dency to expand textbooks with ever more material, excessive explanation, and
more and more bells and whistles simply gets in the way of the student’s under-
standing of the basic material.
The exercises differ widely in level of abstraction and level of diﬃculty. They
vary from the simple to the quite diﬃcult and from the computational to the the-
oretical. There are exercises that ask students to prove something or to construct
an example with certain properties. There are exercises that ask students to apply
theoretical material to help do a computation or to solve a practical problem. Each
section contains a number of examples designed to illustrate the material of the
section and to teach students how to approach the exercises for that section. The
text uses the following convention when referring to exercises: Exercise 1.1.5 is the
fifth exercise in Exercise Set 1.1.
This text, in its various incarnations, has been used by the author and his
colleagues for several years at the University of Utah. Each use has led to improve-
ments, additions, and corrections.