分析 第1卷

　　The present book strives for clarity and transparency. Rightfrom the begin-ning， it requires from the reader a willingness todeal with abstract concepts， as well as a considerable measure ofself-initiative. For these e&，rts， the reader will be richlyrewarded in his or her mathematical thinking abilities， and willpossess the foundation needed for a deeper penetration intomathematics and its applications.
　　This book is the first volume of a three volume introduction toanalysis. It de- veloped from. courses that the authors have taughtover the last twenty six years at the Universities of Bochum， Kiel，Zurich， Basel and Kassel. Since we hope that this book will be usedalso for self-study and supplementary reading， we have included farmore material than can be covered in a three semester sequence.This allows us to provide a wide overview of the subject and topresent the many beautiful and important applications of thetheory. We also demonstrate that mathematics possesses， not onlyelegance and inner beauty， but also provides efficient methods forthe solution of concrete problems.

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