For over three decades, this bestselling classic has been used by thousands of students in the United States and abroad as an essential textbook for a transition course from calculus to analysis. It has proven to be very helpful for mathematics students who have no prior experience with rigorous proofs. The friendly style demystifies the art of writing proofs while carefully examining the theoretical foundations of analysis. The proofs are presented in full, and the large number of well-chosen examples and exercises range from routine to challenging. The second edition maintains the book's clear and concise style, enhances discussions, and provides straightforward, well-motivated proofs. New topics include materials on the irrationality of pi, the Baire category theorem, Newton's method, and the secant method, as well as continuous, nowhere differentiable functions.