Elementary real analysis

✍ Scribed by Thomson B., Bruckner J., Bruckner A.

• Publisher: PH
• Year: 2001
• Tongue: English
• Leaves: 753
• Category: Library

✦ Synopsis

Elementary Real Analysis is written in a rigorous, yet reader friendly style with motivational and historical material that emphasizes the “big picture” and makes proofs seem natural rather than mysterious. Introduces key concepts such as point set theory, uniform continuity of functions and uniform convergence of sequences of functions. Covers metric spaces. Ideal for readers interested in mathematics, particularly in advanced calculus and real analysis.

✦ Table of Contents

Contents......Page 4
Preface......Page 13
Introduction......Page 19
The Real Number System......Page 20
Algebraic Structure......Page 23
Order Structure......Page 26
Bounds......Page 27
Sups And Infs......Page 28
The Archimedean Property......Page 31
Inductive Property Of In......Page 33
The Rational Numbers Are Dense......Page 34
The Metric Structure Of R......Page 36
Challenging Problems For Chapter 1......Page 39
Introduction......Page 41
Sequences......Page 43
Sequence Examples......Page 44
Countable Sets......Page 47
Convergence......Page 50
Divergence......Page 55
Boundedness Properties Of Limits......Page 57
Algebra Of Limits......Page 59
Order Properties Of Limits......Page 65
Monotone Convergence Criterion......Page 70
Examples Of Limits......Page 74
Subsequences......Page 79
Cauchy Convergence Criterion......Page 83
Upper And Lower Limits......Page 86
Challenging Problems For Chapter 2......Page 92
Introduction......Page 95
Finite Sums......Page 96
Infinite Unordered Sums......Page 102
Cauchy Criterion......Page 104
Ordered Sums: Series......Page 108
Properties......Page 109
Special Series......Page 110
Criteria For Convergence......Page 116
Cauchy Criterion......Page 117
Absolute Convergence......Page 118
Trivial Test......Page 122
Direct Comparison Tests......Page 123
Limit Comparison Tests......Page 125
Ratio Comparison Test......Page 126
D'alembert's Ratio Test......Page 127
Cauchy's Root Test......Page 129
Cauchy's Condensation Test......Page 130
Integral Test......Page 132
Kummer's Tests......Page 133
Gauss's Ratio Test......Page 136
Alternating Series Test......Page 139
Dirichlet's Test......Page 140
Abel's Test......Page 141
Rearrangements......Page 147
Unconditional Convergence......Page 148
Conditional Convergence......Page 149
Comparison Of I=1ai And Iin Ai......Page 151
Products Of Series......Page 153
Products Of Absolutely Convergent Series......Page 156
Products Of Nonabsolutely Convergent Series......Page 157
Summability Methods......Page 159
Cesàro's Method......Page 160
Abel's Method......Page 162
More On Infinite Sums......Page 166
Infinite Products......Page 168
Challenging Problems For Chapter 3......Page 172
Introduction......Page 176
Interior Points......Page 177
Points Of Accumulation......Page 179
Boundary Points......Page 180
Sets......Page 183
Closed Sets......Page 184
Open Sets......Page 185
Elementary Topology......Page 191
Compactness Arguments......Page 194
Bolzano-weierstrass Property......Page 196
Cantor's Intersection Property......Page 197
Cousin's Property......Page 199
Heine-borel Property......Page 200
Compact Sets......Page 204
Countable Sets......Page 207
Challenging Problems For Chapter 4......Page 208
Limits (- Definition)......Page 211
Limits (sequential Definition)......Page 215
Limits (mapping Definition)......Page 218
One-sided Limits......Page 219
Infinite Limits......Page 221
Properties Of Limits......Page 222
Boundedness Of Limits......Page 223
Algebra Of Limits......Page 225
Order Properties......Page 228
Composition Of Functions......Page 231
Examples......Page 233
Limits Superior And Inferior......Page 240
How To Define Continuity......Page 241
Continuity At A Point......Page 245
Continuity At An Arbitrary Point......Page 248
Continuity On A Set......Page 250
Properties Of Continuous Functions......Page 253
Uniform Continuity......Page 254
Extremal Properties......Page 258
Darboux Property......Page 259
Types Of Discontinuity......Page 261
Monotonic Functions......Page 263
How Many Points Of Discontinuity?......Page 267
Challenging Problems For Chapter 5......Page 269
Dense Sets......Page 271
Nowhere Dense Sets......Page 273
A Two-player Game......Page 275
The Baire Category Theorem......Page 277
Uniform Boundedness......Page 278
Construction Of The Cantor Ternary Set......Page 280
An Arithmetic Construction Of K......Page 283
The Cantor Function......Page 285
Sets Of Type G......Page 287
Sets Of Type F......Page 289
Oscillation And Continuity......Page 291
Oscillation Of A Function......Page 292
The Set Of Continuity Points......Page 295
Sets Of Measure Zero......Page 297
Challenging Problems For Chapter 6......Page 303
The Derivative......Page 304
Definition Of The Derivative......Page 305
Differentiability And Continuity......Page 310
The Derivative As A Magnification......Page 311
Computations Of Derivatives......Page 312
Algebraic Rules......Page 313
The Chain Rule......Page 316
Inverse Functions......Page 320
The Power Rule......Page 321
Continuity Of The Derivative?......Page 323
Local Extrema......Page 325
Mean Value Theorem......Page 327
Rolle's Theorem......Page 328
Mean Value Theorem......Page 330
Cauchy's Mean Value Theorem......Page 332
Monotonicity......Page 333
Dini Derivates......Page 336
The Darboux Property Of The Derivative......Page 340
Convexity......Page 343
L'hôpital's Rule......Page 348
L'hôpital's Rule: 00 Form......Page 350
L'hôpital's Rule As X......Page 352
L'hôpital's Rule: Form......Page 354
Taylor Polynomials......Page 357
Challenging Problems For Chapter 7......Page 361
Introduction......Page 364
Cauchy's First Method......Page 367
Scope Of Cauchy's First Method......Page 369
Properties Of The Integral......Page 372
Cauchy's Second Method......Page 377
Cauchy's Second Method (continued)......Page 380
The Riemann Integral......Page 382
Some Examples......Page 384
Riemann's Criteria......Page 386
Lebesgue's Criterion......Page 388
What Functions Are Riemann Integrable?......Page 391
Properties Of The Riemann Integral......Page 392
The Improper Riemann Integral......Page 396
More On The Fundamental Theorem Of Calculus......Page 398
Challenging Problems For Chapter 8......Page 400
Introduction......Page 402
Pointwise Limits......Page 403
Uniform Limits......Page 409
The Cauchy Criterion......Page 412
Weierstrass M-test......Page 414
Abel's Test For Uniform Convergence......Page 416
Uniform Convergence And Continuity......Page 422
Dini's Theorem......Page 423
Sequences Of Continuous Functions......Page 426
Sequences Of Riemann Integrable Functions......Page 428
Sequences Of Improper Integrals......Page 430
Uniform Convergence And Derivatives......Page 433
Limits Of Discontinuous Derivatives......Page 435
Pompeiu's Function......Page 437
Continuity And Pointwise Limits......Page 440
Challenging Problems For Chapter 9......Page 443
Introduction......Page 444
Power Series: Convergence......Page 445
Uniform Convergence......Page 450
Continuity Of Power Series......Page 453
Integration Of Power Series......Page 454
Differentiation Of Power Series......Page 455
Power Series Representations......Page 458
The Taylor Series......Page 461
Representing A Function By A Taylor Series......Page 462
Analytic Functions......Page 465
Products Of Power Series......Page 467
Quotients Of Power Series......Page 468
Composition Of Power Series......Page 470
Trigonometric Series......Page 471
Uniform Convergence Of Trigonometric Series......Page 472
Fourier Series......Page 473
Convergence Of Fourier Series......Page 474
Weierstrass Approximation Theorem......Page 478
The Algebraic Structure Of Rn......Page 480
The Metric Structure Of Rn......Page 482
Elementary Topology Of Rn......Page 486
Sequences In Rn......Page 488
Functions From Rnr......Page 493
Functions From Rnrm......Page 495
Definition......Page 498
Coordinate-wise Convergence......Page 501
Algebraic Properties......Page 503
Continuity Of Functions From Rn To Rm......Page 504
Compact Sets In Rn......Page 507
Continuous Functions On Compact Sets......Page 508
Additional Remarks......Page 509
Introduction......Page 513
Partial And Directional Derivatives......Page 514
Partial Derivatives......Page 515
Directional Derivatives......Page 518
Cross Partials......Page 519
Integrals Depending On A Parameter......Page 524
Differentiable Functions......Page 528
Approximation By Linear Functions......Page 529
Definition Of Differentiability......Page 530
Differentiability And Continuity......Page 534
Directional Derivatives......Page 535
An Example......Page 537
Sufficient Conditions For Differentiability......Page 539
The Differential......Page 541
Preliminary Discussion......Page 544
Informal Proof Of A Chain Rule......Page 548
Notation Of Chain Rules......Page 549
Proofs Of Chain Rules (i)......Page 551
Mean Value Theorem......Page 553
Proofs Of Chain Rules (ii)......Page 554
Higher Derivatives......Page 556
Implicit Function Theorems......Page 559
One-variable Case......Page 560
Several-variable Case......Page 563
Simultaneous Equations......Page 567
Inverse Function Theorem......Page 571
Functions From Rrm......Page 574
Functions From Rnrm......Page 577
Review Of Differentials And Derivatives......Page 578
Definition Of The Derivative......Page 580
Jacobians......Page 582
Chain Rules......Page 585
Proof Of Chain Rule......Page 587
Introduction......Page 591
Metric Spaces---specific Examples......Page 593
Sequence Spaces......Page 598
Function Spaces......Page 600
Convergence......Page 603
Sets In A Metric Space......Page 607
Functions......Page 615
Continuity......Page 617
Homeomorphisms......Page 622
Isometries......Page 628
Separable Spaces......Page 631
Complete Spaces......Page 634
Completeness Proofs......Page 635
Cantor Intersection Property......Page 637
Completion Of A Metric Space......Page 638
Contraction Maps......Page 641
Applications Of Contraction Maps (i)......Page 648
Applications Of Contraction Maps (ii)......Page 651
Systems Of Equations (example 13.79 Revisited)......Page 652
Infinite Systems (example 13.80 Revisited)......Page 653
Integral Equations (example 13.81 Revisited)......Page 655
Picard's Theorem (example 13.82 Revisited)......Page 656
Compactness......Page 658
The Bolzano-weierstrass Property......Page 659
Continuous Functions On Compact Sets......Page 662
The Heine-borel Property......Page 664
Total Boundedness......Page 666
Compact Sets In C[a,b]......Page 669
Peano's Theorem......Page 674
Baire Category Theorem......Page 677
Nowhere Dense Sets......Page 678
The Baire Category Theorem......Page 681
Functions Whose Graphs ``cross No Lines''......Page 684
Nowhere Monotonic Functions......Page 688
Continuous Nowhere Differentiable Functions......Page 689
Cantor Sets......Page 690
Challenging Problems For Chapter 13......Page 692
Set Notation......Page 696
Function Notation......Page 700
Why Proofs?......Page 706
Indirect Proof......Page 708
Contraposition......Page 709
Counterexamples......Page 710
Induction......Page 711
Quantifiers......Page 714
Appendix B: Hints For Selected Exercises......Page 717
Subject Index......Page 734