From Numbers to Analysis
Starting with the Zermelo-Fraenhel axiomatic set theory, this book gives a self-contained, step-by-step construction of real and complex numbers. The basic properties of real and complex numbers are developed, including a proof of the Fundamental Theorem of Algebra. Historical notes outline the evolution of the number systems and alert readers to the fact that polished mathematical concepts, as presented in lectures and books, are the culmination of the efforts of great minds over the years. The text also includes short life sketches of some of the contributing mathematicians. The book provides the logical foundation of Analysis and gives a basis to Abstract Algebra. It complements those books on real analysis which begin with axiomatic definitions of real numbers.The book can be used in various ways: as a textbook for a one semester course on the foundations of analysis for post-calculus students; for a seminar course; or self-study by school and college teachers.
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PROPERTIES OF REAL NUMBERS
SUGGESTIONS FOR FURTHER READING
absolute value addition and multiplication algebraic Archimedean property axiom A(1 axiom of choice axiomatic set theory binary operation called Cantor Cauchy complete Cauchy sequence choose Clearly commutative complex numbers Consider the set constructed contradiction Corollary countably infinite defined Definition denote easy to check elements example Exercise exists field of rationals finite set following hold function given glb(S Hence identity implies induction infinite set integer isomorphic Let F limit point lub(A lub(S mathematicians mathematics metric space naive set theory nested interval nonempty set nonempty subset Note one-one map open interval open set ordered field ordered integral domain polynomial positive integer prime Proof Prove the following r e Q rational numbers real numbers sequence an}n-1 Similarly Suppose symbols theorem unique upper bound