Table of contents for Closer and closer : introducing real analysis / Carol Schumacher.

Bibliographic record and links to related information available from the Library of Congress catalog.

Note: Contents data are machine generated based on pre-publication provided by the publisher. Contents may have variations from the printed book or be incomplete or contain other coding.


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Contents 
I Central Ideas 7
Preliminary Remarks .......................... 8
WhatisAnalysis? .......................... 8
TheRoleofAbstraction ....................... 10
AThoughtExperiment ....................... 11
0 Basic Building Blocks 13 
0.1 SetsandSetNotation ........................ 13
TotallyOrderedSets ......................... 15
CollectionsofSets|indexingsets.................. 16
SetOperations ............................ 18
Problems .................................. 21
0.2 Functions ............................... 22
InverseFunctions ........................... 24
ImagesandInverseImages...................... 25
Problems .................................. 28
0.3 The Natural Numbers, the Integers and their Properties . . . . . 31 
MathematicalInduction ....................... 32 
0.4 Sequences ............................... 33
Subsequences ............................. 35
Problems .................................. 39
1 The Real Numbers 40 
1.1 ConstructingtheAxioms ...................... 40
1.2 Arithmetic............................... 40
The\WishList¿ ........................... 42
Problems .................................. 45
1.3 Order ................................. 46
OrderandArithmetic ........................ 47
AbsoluteValues............................ 48
Backtothe\WishList¿ ....................... 49
Problems .................................. 53
1.4 TheLeastUpperBoundAxiom ................... 56
Problems .................................. 64
1 
2 Measuring Distances 66 
2.1 MetricSpaces ............................. 66
2.2 The Euclidean Metric on Rn .................... 68
TheCauchy-SchwarzInequality ................... 70
Problems .................................. 71
3 Sets and Limits 74 
3.1 OpenSets ............................... 74
BoundednessinMetricSpaces.................... 76
Problems .................................. 79
3.2 Convergence of Sequences: Thinking Intuitively . . . . . . . . . . 83 
3.3 ConvergenceofSequences ...................... 84
Problems .................................. 87
3.4 Sequences in R ............................ 88 
SequenceConvergenceandOrder .................. 88 
SequenceConvergenceandArithmetic . . . . . . . . . . . . . . . 90 
Problems .................................. 93 
3.5 LimitPoints.............................. 95
Problems .................................. 97
3.6 ClosedSets .............................. 98
Problems .................................. 99
3.7 OpenSets,ClosedSetsandtheClosureofaSet. . . . . . . . . . 100 
Problems .................................. 104 
4 Continuity 106 
4.1 ThinkingIntuitively ......................... 106
4.2 LimitofaFunctionataPoint.................... 107
Problems .................................. 111
4.3 ContinuousFunctions ........................ 113
Problems .................................. 115
4.4 UniformContinuity.......................... 118
Problems .................................. 119
5 Real-Valued Functions 121 
5.1 Limits,Continuity,andOrder .................... 122
SomeUsefulSpecialCases...................... 124
Problems .................................. 126
5.2 One-sidedlimits ........................... 127
Problems .................................. 128
5.3 Limits,Continuity,andArithmetic . . . . . . . . . . . . . . . . . 129 
Problems .................................. 131 
2 
6 Completeness 133 
6.1 CauchySequences .......................... 133
Problems .................................. 135
6.2 CompleteMetricSpaces ....................... 136
Problems .................................. 138
7 Compactness 139 
7.1 CompactSets ............................. 139
Problems .................................. 145
7.2 ContinuityandCompactness .................... 148
Problems .................................. 149
7.3 Compactness in Rn .......................... 151
Problems .................................. 155
8 Connectedness 156 
8.1 TheIntermediateValueTheorem .................. 156
Problems .................................. 158
8.2 ConnectedSets ............................ 159
Problems .................................. 162
9 Di«erentiation of Functions of One Real Variable 164 
9.1 RegardingDomains.......................... 164
9.2 TheDerivative ............................ 166
Problems .................................. 171
9.3 What Does the Derivative Tell us about the Function? . . . . . . 175 
9.4 ProvingtheMeanValueTheorem ................. 176
Problems .................................. 180
9.5 Monotonicity and the Mean Value Theorem . . . . . . . . . . . . 183 
Problems .................................. 186 
9.6 InverseFunctions ........................... 188
Problems .................................. 190
9.7 Polynomial Approximation and Taylor's Theorem . . . . . . . . . 191 
Problems .................................. 195 
10 Iteration and the Contraction Mapping Theorem 196 
10.1IterationandFixedPoints...................... 196
AttractorsandRepellors....................... 201
Problems .................................. 203
10.2 TheContractionMappingTheorem . . . . . . . . . . . . . . . . 205 
WhyYouShouldCareAboutFixedPoints . . . . . . . . . . . . 207 
Problems .................................. 209 
10.3 More on Finding Attracting Fixed Points . . . . . . . . . . . . . 213 
Problems .................................. 216 
3 
11 The Riemann Integral 217 
11.1Whatisarea? ............................. 217
11.2TheRiemannIntegral ........................ 218
Problems .................................. 224
11.3 Arithmetic,OrderandtheIntegral . . . . . . . . . . . . . . . . . 226 
Problems .................................. 227 
11.4FamiliesofRiemannSums...................... 228 
The Riemann \Envelope": Upper and Lower Sums . . . . . . . . 228 
Re»nements .............................. 229 
Cauchy Criteria for the Existence of the Integral . . . . . . . . . 230 
Problems .................................. 235 
11.5ExistenceoftheIntegral ....................... 236
Problems .................................. 240
11.6 TheFundamentalTheoremofCalculus. . . . . . . . . . . . . . . 243 
Problems .................................. 244 
12 Sequences of Functions 245 
12.1Pointwiseconvergence ........................ 245
12.2UniformConvergence ........................ 248
Problems .................................. 251
12.3SeriesofFunctions .......................... 255
Problems .................................. 258
12.4 InterchangeofLimitOperations . . . . . . . . . . . . . . . . . . 259 
Problems .................................. 265 
13 Di«erentiating f : Rn Rm 266 
!
13.1Whatarewestudying?........................ 266
Problems .................................. 269
13.2ThinkingIntuitively ......................... 270
TangentPlanes ............................ 270
13.3AnalysisinLinearSpaces ...................... 272
Lineartransformations ........................ 274
LinearAlgebraandAnalysis..................... 276
Problems .................................. 280
13.4 Local Linear Approximation for Functions of Several Variables . 282 
Connections|Total and Partial Derivatives . . . . . . . . . . . . 283 
Problems .................................. 292 
13.5 The Mean Value Theorem for Functions of Several Variables . . . 298 
Problems .................................. 301 
II Excursions 302 
1 Truth and Provability 303 
2 Number Properties 305 
4 
3 Exponents 308 
3.1 IntegerandRationalPowers..................... 308
PositiveIntegerPowers ....................... 308
3.2 IrrationalPowers ........................... 313
4 Sequences in R and Rn 316 
4.1 Sequence Convergence in R and Rn ................. 316
4.2 Epsilonics|PlayingtheGame ................... 319 
VoodooMathematics? ........................ 319 
ScratchWork|DevisingaStrategy. . . . . . . . . . . . . . . . . 321 
Problems .................................. 323 
4.3 In»niteLimits............................. 324
4.4 SomeImportantSpecialSequences . . . . . . . . . . . . . . . . . 324 
5 Limits of Functions from R to R 327 
5.1 ExampleProofs............................ 327
5.2 Epsilonics|SomeGeneralPrinciples ................ 329
Problems .................................. 331
6 Doubly Indexed Sequences 333 
6.1 DoubleSequencesandConvergence. . . . . . . . . . . . . . . . . 334 
7 Subsequences and Convergence 338 
7.1 SubsequentialLimits ......................... 338
7.2 LimitsSuperiorandInferior ..................... 339
8 Series of Real Numbers 342 
8.1 De»nitionandBasicProperties ................... 342 
GeometricSeries ........................... 343 
Cauchy Criterion for Series Convergence . . . . . . . . . . . . . . 344 
NthTermTest ............................ 344 
Absolute vs. Conditional Convergence . . . . . . . . . . . . . . . 346 
Problems .................................. 348 
8.2 ComparingSeries ........................... 350
Problems .................................. 352
8.3 RelativesoftheGeometricSeries . . . . . . . . . . . . . . . . . . 353 
ComparingtheRootandtheRatioTests . . . . . . . . . . . . . 354 
Problems .................................. 356 
8.4 RearrangingtheTermsofaSeries . . . . . . . . . . . . . . . . . 358 
Problems .................................. 363 
8.5 MultiplyingSeries .......................... 364
9 Probing the De»nition of the Riemann Integral 367 
9.1 RegularRiemannSums ....................... 367
9.2 WhytheGenerality? ......................... 369
Problems .................................. 371
5 
10 Power Series 372 
10.1 De»nitions and Convergence of Power Series . . . . . . . . . . . . 372 
Problems .................................. 376 
10.2 Integration and Di«erentiation of Power Series . . . . . . . . . . 377 
Problems .................................. 379 
10.3TaylorSeries ............................. 380
Problems .................................. 382
11 Everywhere Continuous, Nowhere Di«erentiable 383 
11.1Introduction.............................. 383
11.2Constructingthefunction ...................... 384
12 Newton's Method 389 
12.1SettingtheStage ........................... 389
12.2IteratingtheNewtonFunction ................... 392
Problems .................................. 393
12.3 Experimenting with Newton's Method . . . . . . . . . . . . . . . 394 
12.4 On Choosing x0 ........................... 395
12.5ConvergenceRate .......................... 396
Problems .................................. 398
13 The Implicit Function Theorem 399 
13.1SolvingSystemsofEquations .................... 399
13.2TheImplicitFunctionTheorem ................... 402
WhatonEarth?!? .......................... 403
PropertiesoftheSolutionFunction................. 406
Problems .................................. 408
13.3 Connections|Quasi-Newton's methods . . . . . . . . . . . . . . 412 
13.4TheInverseFunctionTheorem ................... 416
Problems .................................. 417
14 Spaces of Continuous Functions 418 
14.1 The metric space C(K) ....................... 420
14.2 Compactness in C(K) ........................ 422
15 Solutions to Di«erential Equations 424 
15.1De»nitionsandMotivation ..................... 424
WhyExistence?WhyUniqueness? ................. 426
15.2 Picard Iteration route to Existence and Uniqueness . . . . . . . . 427 
Problems .................................. 430 
15.3SystemsofEquations......................... 432
Problems .................................. 435

Library of Congress Subject Headings for this publication:

Functions.
Mathematical analysis.