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131104  eng 
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a 9783319015774

100 
1 

a Stillwell, John

245 
0 
0 
a The Real Numbers
h Elektronische Ressource
b An Introduction to Set Theory and Analysis
c by John Stillwell

250 


a 1st ed. 2013

260 


a Cham
b Springer International Publishing
c 2013, 2013

300 


a XVI, 244 p. 62 illus
b online resource

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0 

a The 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

653 


a History of Mathematical Sciences

653 


a Mathematical logic

653 


a Functions of real variables

653 


a History

653 


a Mathematical Logic and Foundations

653 


a Real Functions

653 


a Mathematics

710 
2 

a SpringerLink (Online service)

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0 
7 
a eng
2 ISO 6392

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b Springer
a Springer eBooks 2005

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0 

a Undergraduate Texts in Mathematics

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u https://doi.org/10.1007/9783319015774?nosfx=y
x Verlag
3 Volltext

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0 

a 515.8

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a While 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 settheoretic aspects of analysis, this text makes the best of two worlds: it combines a downtoearth 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
