The Real Numbers and Real Analysis

577 Pages·2011·4.82 MB·English

The Real Numbers and Real Analysis

Ethan D. Bloch
The Real Numbers
and Real Analysis
Ethan D. Bloch
Mathematics Department
Bard College
Annandale-on-Hudson, NY 12504
USA
[email protected]

ISBN 978-0-387-72176-7 e-ISBN 978-0-387-72177-4
DOI 10.1007/978-0-387-72177-4
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2011928556
Mathematics Subject Classification (2010): 26-01
© Springer Science+Business Media, LLC 2011
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DedicatedtomytwowonderfulchildrenGilNehemyaandAdaHaviva,
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Contents
Preface............................................................ xi
TotheStudent ..................................................... xxi
TotheInstructor ...................................................xxvii
1 ConstructionoftheRealNumbers ................................ 1
1.1 Introduction ................................................ 1
1.2 Entry1:AxiomsfortheNaturalNumbers ...................... 2
1.3 ConstructingtheIntegers ..................................... 11
1.4 Entry2:AxiomsfortheIntegers .............................. 19
1.5 ConstructingtheRationalNumbers ........................... 27
1.6 DedekindCuts............................................. 33
1.7 ConstructingtheRealNumbers................................ 41
1.8 HistoricalRemarks .......................................... 51
2 PropertiesoftheRealNumbers................................... 61
2.1 Introduction ................................................ 61
2.2 Entry3:AxiomsfortheRealNumbers......................... 62
2.3 AlgebraicPropertiesoftheRealNumbers ...................... 65
2.4 Finding the Natural Numbers, the Integers and the Rational
NumbersintheRealNumbers................................ 75
2.5 InductionandRecursioninPractice ........................... 83
2.6 TheLeastUpperBoundPropertyandItsConsequences .......... 96
2.7 UniquenessoftheRealNumbers.............................. 107
2.8 DecimalExpansionofRealNumbers.......................... 113
2.9 HistoricalRemarks ......................................... 128
3 LimitsandContinuity .......................................... 129
3.1 Introduction ............................................... 129
3.2 LimitsofFunctions......................................... 129
viii Contents
3.3 Continuity ................................................ 146
3.4 UniformContinuity......................................... 156
3.5 TwoImportantTheorems .................................... 163
3.6 HistoricalRemarks ..........................................171
4 Differentiation..................................................181
4.1 Introduction ................................................181
4.2 TheDerivative ..............................................181
4.3 ComputingDerivatives...................................... 192
4.4 TheMeanValueTheorem ................................... 198
4.5 IncreasingandDecreasingFunctions,PartI:LocalandGlobal
Extrema .................................................. 207
4.6 IncreasingandDecreasingFunctions,PartII:FurtherTopics ...... 215
4.7 HistoricalRemarks ......................................... 225
5 Integration .....................................................231
5.1 Introduction ................................................231
5.2 TheRiemannIntegral ........................................231
5.3 ElementaryPropertiesoftheRiemannIntegral .................. 242
5.4 UpperSumsandLowerSums ................................ 247
5.5 FurtherPropertiesoftheRiemannIntegral ..................... 258
5.6 FundamentalTheoremofCalculus ............................ 267
5.7 ComputingAntiderivatives................................... 277
5.8 Lebesgue’sTheorem........................................ 283
5.9 AreaandArcLength........................................ 293
5.10 HistoricalRemarks ......................................... 312
6 LimitstoInfinity................................................321
6.1 Introduction ................................................321
6.2 LimitstoInfinity ........................................... 322
6.3 ComputingLimitstoInfinity ..................................331
6.4 ImproperIntegrals...........................................341
6.5 HistoricalRemarks ......................................... 354
7 TranscendentalFunctions....................................... 357
7.1 Introduction ............................................... 357
7.2 LogarithmicandExponentialFunctions........................ 358
7.3 TrigonometricFunctions .................................... 369
7.4 Moreaboutπ .............................................. 379
7.5 HistoricalRemarks ..........................................391
8 Sequences ..................................................... 399
8.1 Introduction ............................................... 399
8.2 Sequences................................................. 399
8.3 ThreeImportantTheorems................................... 412
8.4 ApplicationsofSequences ................................... 423
Contents ix
8.5 HistoricalRemarks ......................................... 439
9 Series......................................................... 443
9.1 Introduction ............................................... 443
9.2 Series .................................................... 443
9.3 ConvergenceTests...........................................451
9.4 AbsoluteConvergenceandConditionalConvergence............. 459
9.5 PowerSeriesasFunctions ................................... 473
9.6 HistoricalRemarks ......................................... 482
10 SequencesandSeriesofFunctions ............................... 489
10.1 Introduction ............................................... 489
10.2 SequencesofFunctions ..................................... 489
10.3 SeriesofFunctions ......................................... 502
10.4 FunctionsasPowerSeries ................................... 509
10.5 AContinuousbutNowhereDifferentiableFunction.............. 527
10.6 HistoricalRemarks ......................................... 534
Bibliography....................................................... 539
Index ............................................................. 545