This book is a one-semester text for an introduction to real analysis. The author's primary aims are to develop ideas already familiar from elementary calculus in a rigorous manner and to help students deeply understand some basic but crucial mathematical ideas, and to see how definitions, proofs, examples, and other forms of mathematical "apparatus" work together to create a unified theory. A key feature of the book is that it includes substantial treatment of some foundational material, including general theory of functions, sets, cardinality, and basic proof techniques.

Reviews

This is a textbook designed to teach students who are new to analysis what it’s all about. … The path Zorn takes is based on several very reasonable principles. These include: building on calculus basics; focusing on mathematical proof, structure and language; staying with the basics; offering many examples and many solved exercises; and gradually increasing technical sophistication. … There are plenty of exercises. They tend to follow a pattern where an exercise that is not completely straightforward is broken into multiple parts to guide the student to a solution.

-- Bill Satzer, MAA Reviews, June 2010

Contents

Preface

1 Preliminaries: Numbers, Sets, Proofs, and Bounds

Numbers 101: The Very Basics

Sets 101: Getting Started

Sets 102: The Idea of a Function

Proofs 101: Proofs and Proof-Writing

Types of Proof

Sets 103: Finite and Infinite Sets; Cardinality

Numbers 102: Absolute Values

Bounds

Numbers 103: Completeness

2 Sequences and Series

SequencesandConvergence

WorkingwithSequences

Subsequences

CauchySequences

Series 101: Basic Ideas

Series 102: Testing for Convergence and Estimating Limits

Limsupandliminf:AGuidedDiscovery

3 Limits and Continuity

LimitsofFunctions

Continuous Functions

WhyContinuityMatters:ValueTheorems

Unifority

4 Derivatives

DefiningtheDerivative

CalculatingDerivatives

TheMeanValueTheorem

SequencesofFunctions

5 Integrals

The Riemann Integral: Definition and Examples

Propertiesof the Integral

Integrability

Some Fundamental Theorems

Solutions

Author Bio

Paul Zorn was born in India and completed his primary and secondary schooling there. He did his undergraduate work at Washington University in St. Louis and his Ph.D., in complex analysis, at the University of Washington, Seattle. Since 1981 he has been on the mathematics faculty at St. Olaf College, in Northfield, Minnesota, where he now chairs the Department of Mathematics, Statistics, and Computer Science.