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000706852 010__ $$a  2013946655
000706852 019__ $$a857977710
000706852 020__ $$a9783319015767$$qhardcover$$qalkaline paper
000706852 020__ $$a3319015761$$qhardcover$$qalkaline paper
000706852 035__ $$a(OCoLC)ocn856738499
000706852 040__ $$aYDXCP$$beng$$erda$$cYDXCP$$dGZM$$dBTCTA$$dOHX$$dAMH$$dOCLCF
000706852 049__ $$aISEA
000706852 050_4 $$aQA248$$b.S777 2013
000706852 1001_ $$aStillwell, John,$$eauthor.
000706852 24514 $$aThe real numbers :$$ban introduction to set theory and analysis /$$cJohn Stillwell.
000706852 264_1 $$aCham [Switzerland] ;$$aNew York :$$bSpringer,$$c[2013]
000706852 264_4 $$c©2013
000706852 300__ $$axvi, 244 pages :$$billustrations ;$$c25 cm.
000706852 336__ $$atext$$btxt$$2rdacontent
000706852 337__ $$aunmediated$$bn$$2rdamedia
000706852 338__ $$avolume$$bnc$$2rdacarrier
000706852 4901_ $$aUndergraduate texts in mathematics,$$x0172-6056
000706852 504__ $$aIncludes bibliographical references (pages 225-229) and index.
000706852 5050_ $$aThe fundamental questions -- From discrete to continuous -- Infinite sets -- Functions and limits -- Open sets and continuity -- Ordinals -- The axiom of choice -- Borel sets -- Measure theory -- Reflections.
000706852 520__ $$aWhile most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory"uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis"the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor-Schröder-Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.
000706852 650_0 $$aSet theory.
000706852 650_0 $$aMathematical analysis.
000706852 830_0 $$aUndergraduate texts in mathematics.
000706852 85200 $$bgen$$hQA248$$i.S777$$i2013
000706852 909CO $$ooai:library.usi.edu:706852$$pGLOBAL_SET
000706852 980__ $$aBIB
000706852 980__ $$aBOOK