Elementary Real Analysis

"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. 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 Written in a reader-friendly style with historical material emphasizing the "big picture", this text makes proofs seem natural rather than mysterious. It introduces key concepts such as point set theory, uniform continuity of functions and uniform convergence of sequences of functions.