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020 _a9783319015774
_9978-3-319-01577-4
024 7 _a10.1007/978-3-319-01577-4
_2doi
050 4 _aQA331.5
072 7 _aPBKB
_2bicssc
072 7 _aMAT034000
_2bisacsh
072 7 _aMAT037000
_2bisacsh
082 0 4 _a515.8
_223
100 1 _aStillwell, John.
_eauthor.
245 1 4 _aThe Real Numbers
_h[electronic resource] :
_bAn Introduction to Set Theory and Analysis /
_cby John Stillwell.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2013.
300 _aXVI, 244 p. 62 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aUndergraduate Texts in Mathematics,
_x0172-6056
505 0 _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 -- Bibliography -- Index.
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.
650 0 _aMathematics.
650 0 _aLogic, Symbolic and mathematical.
650 1 4 _aMathematics.
650 2 4 _aReal Functions.
650 2 4 _aMathematical Logic and Foundations.
650 2 4 _aHistory of Mathematical Sciences.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783319015767
830 0 _aUndergraduate Texts in Mathematics,
_x0172-6056
856 4 0 _uhttp://dx.doi.org/10.1007/978-3-319-01577-4
912 _aZDB-2-SMA
999 _c96551
_d96551