分析

Preface

Chapter Ⅰ Foundations

1 Fundamentals of Logic

2 Sets

Elementary Facts

The Power Set

Complement, Intersection and Union

Products

Families of Sets

3 Functions，

Simple Examples

Composition of Functions

Commutative Diagrams

Injections, Surjections and Bijections

Inverse Functions

Set Valued Functions

4 Relations and Operations

Equivalence Relations

Order Relations

Operations

5 The Natural Numbers

The Peano Axioms

The Arithmetic of Natural Numbers

The Division Algorithm

The Induction Principle

Recursive Definitions

6 Countability

Permutations

Equinumerous Sets

Countable Sets

Infinite Products

7 Groups and Homomorphisms

Groups

Subgroups

Cosets

Homomorphisms

Isomorphisms

8 R.ings, Fields and Polynomials

Rings

The Binomial Theorem

The Multinomial Theorem

Fields

Ordered Fields

Formal Power Series

Polynomials

Polynomial Functions

Division of Polynomiajs

Linear Factors

Polynomials in Several Indeterminates

9 The Rational Numbers

The Integers

The Rational Numbers

Rational Zeros of Polynomials

Square Roots

10 The Real Numbers

Order Completeness

Dedekind's Construction of the Real Numbers

The Natural Order on R

The Extended Number Line

A Characterization of Supremum and Infimum

The Archimedean Property

The Density of the Rational Numbers in R

nth Roots

The Density of the Irrational Numbers in R

Intervals

Chapter Ⅱ Convergence

Chapter Ⅲ Continuous Functions

Chapter Ⅳ Differentiation in One Variable

Chapter Ⅴ Sequences of Functions

Appendix Introduction to Mathematical Logic

Bibliography

Index

The present book strives for clarity and transparency. Right from the begin-ning， it requires from the reader a willingness to deal with abstract concepts， as well as a considerable measure of self-initiative. For these e&，rts， the reader will be richly rewarded in his or her mathematical thinking abilities， and will possess the foundation needed for a deeper penetration into mathematics and its applications.
　　This book is the first volume of a three volume introduction to analysis. It de- veloped from. courses that the authors have taught over the last twenty six years at the Universities of Bochum， Kiel， Zurich， Basel and Kassel. Since we hope that this book will be used also for self-study and supplementary reading， we have included far more material than can be covered in a three semester sequence. This allows us to provide a wide overview of the subject and to present the many beautiful and important applications of the theory. We also demonstrate that mathematics possesses， not only elegance and inner beauty， but also provides efficient methods for the solution of concrete problems.