This book is designed for a first course in real analysis which follows the standard course in elementary calculus. Since many students encounter rigorous mathematical theory for the first time in this course, the authors include such elementary topics as the axioms of algebra and their immediate consequences and proofs of theorems on limits. The pace is deliberately slow, the proofs are detailed. The emphasis of the presentation is on theory, but the books also contains a full treatment (with many illustrative examples and exercises) of the standard topics in infinite series, Fourier series, multidimensional calculus, elements of metric spaces, and vector field theory. There are many problems which require the student to learn techniques of proofs and the standard tools of analysis. -- Back cover.