One Variable Advanced Calculus

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Contents1 Introduction 72 The Real And Complex Numbers 92.1 The Number Line And Algebra Of The Real Numbers . . . . . . . . . . 92.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Set Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.7 Well Ordering Principle And Archimedean Property . . . . . . . . . . . 202.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.9 Completeness of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.10 Existence Of Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.12 The Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Set Theory 353.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 The Schroder Bernstein Theorem . . . . . . . . . . . . . . . . . . . . . . 373.3 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Functions And Sequences 434.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4 The Limit Of A Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 494.5 The Nested Interval Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 524.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.7 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.7.1 Sequential Compactness . . . . . . . . . . . . . . . . . . . . . . . 554.7.2 Closed And Open Sets . . . . . . . . . . . . . . . . . . . . . . . . 554.7.3 Compactness And Open Coverings . . . . . . . . . . . . . . . . . 594.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.9 Cauchy Sequences And Completeness . . . . . . . . . . . . . . . . . . . 624.9.1 Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.9.2 lim sup and lim inf . . . . . . . . . . . . . . . . . . . . . . . . . . 664.9.3 Shrinking Diameters . . . . . . . . . . . . . . . . . . . . . . . . . 694.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
Page 1: One Variable Advanced CalculusKenne
Page 5 and 6: CONTENTS 59 The Riemann And Riemann
Page 7 and 8: Chapter 1IntroductionThe difference
Page 9 and 10: Chapter 2The Real And ComplexNumber
Page 11 and 12: 2.1. THE NUMBER LINE AND ALGEBRA OF
Page 13 and 14: 2.2. EXERCISES 132.2 Exercises1. Co
Page 15 and 16: 2.4. ORDER 15Theorem 2.4.5 The foll
Page 17 and 18: 2.4. ORDER 17Now repeat the argumen
Page 19 and 20: 2.6. THE BINOMIAL THEOREM 1910. Sol
Page 21 and 22: 2.7. WELL ORDERING PRINCIPLE AND AR
Page 23 and 24: 2.7. WELL ORDERING PRINCIPLE AND AR
Page 25 and 26: 2.8. EXERCISES 25In general, the am
Page 27 and 28: 2.9. COMPLETENESS OF R 272.9 Comple
Page 29 and 30: 2.11. EXERCISES 29Proof: Let S deno
Page 31 and 32: 2.12. THE COMPLEX NUMBERS 31The exp
Page 33 and 34: 2.13. EXERCISES 33Proof: First note
Page 35 and 36: Chapter 3Set Theory3.1 Basic Defini
Page 37 and 38: 3.2. THE SCHRODER BERNSTEIN THEOREM
Page 39 and 40: 3.2. THE SCHRODER BERNSTEIN THEOREM
Page 41 and 42: 3.4. EXERCISES 411. x ∼ x for all
Page 43 and 44: Chapter 4Functions And Sequences4.1
Page 45 and 46: 4.1. GENERAL CONSIDERATIONS 45Note
Page 47 and 48: 4.3. EXERCISES 474.3 Exercises1. Le
Page 49 and 50: 4.4. THE LIMIT OF A SEQUENCE 494.4
Page 51 and 52: 4.4. THE LIMIT OF A SEQUENCE 51Thus
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4.6. EXERCISES 53Lemma 4.5.1 Let I
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4.7. COMPACTNESS 554.7 Compactness4
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4.7. COMPACTNESS 57Proof: Is it the
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4.7. COMPACTNESS 59and so by the tr
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4.8. EXERCISES 6118. Show that any
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4.9. CAUCHY SEQUENCES AND COMPLETEN
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4.10. EXERCISES 714. Using Corollar
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4.10. EXERCISES 7327. Give an examp
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Chapter 5Infinite Series Of Numbers
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5.1. BASIC CONSIDERATIONS 77Proof:
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5.1. BASIC CONSIDERATIONS 79This is
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5.1. BASIC CONSIDERATIONS 81enough
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5.3. MORE TESTS FOR CONVERGENCE 834
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5.3. MORE TESTS FOR CONVERGENCE 851
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5.4. DOUBLE SERIES 87Corollary 5.3.
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5.4. DOUBLE SERIES 89Proof: Note th
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5.4. DOUBLE SERIES 91and so∞∑
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5.5. EXERCISES 936. If ∑ ∞n=1 a
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Chapter 6Continuous FunctionsThe co
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values of x, |f (x) − f (x 0 )| >
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99≤ |a| |f (x) − f (y)| + |b| |
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6.1. EQUIVALENT FORMULATIONS OF CON
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6.2. EXERCISES 1034. Let f (x) = 2x
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6.4. THE INTERMEDIATE VALUE THEOREM
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6.4. THE INTERMEDIATE VALUE THEOREM
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6.5. CONNECTED SETS 109Theorem 6.5.
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6.7. UNIFORM CONTINUITY 11115. Let
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6.9. SEQUENCES AND SERIES OF FUNCTI
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6.9. SEQUENCES AND SERIES OF FUNCTI
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6.10. SEQUENCES OF POLYNOMIALS, WEI
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6.10. SEQUENCES OF POLYNOMIALS, WEI
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6.11. ASCOLI ARZELA THEOREM ∗ 121
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6.12. EXERCISES 123a sequence of po
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Chapter 7The DerivativeSome functio
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7.1. LIMIT OF A FUNCTION 127It may
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7.1. LIMIT OF A FUNCTION 129Proof:
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7.3. THE DEFINITION OF THE DERIVATI
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7.3. THE DEFINITION OF THE DERIVATI
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7.4. CONTINUOUS AND NOWHERE DIFFERE
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7.4. CONTINUOUS AND NOWHERE DIFFERE
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7.5. FINDING THE DERIVATIVE 139= f
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7.7. EXERCISES 141Theorem 7.6.2 Sup
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7.8. MEAN VALUE THEOREM 143every co
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7.8. MEAN VALUE THEOREM 145Proof: I
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7.10. DERIVATIVES OF INVERSE FUNCTI
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7.11. DERIVATIVES AND LIMITS OF SEQ
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7.12. EXERCISES 1514. Using the res
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Chapter 8Power Series8.1 Functions
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8.2. OPERATIONS ON POWER SERIES 155
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8.3. THE SPECIAL FUNCTIONS OF ELEME
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8.4. THE BINOMIAL THEOREM 165and so
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8.5. EXERCISES 1678. Suppose f (x +
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8.6.L’HÔPITAL’S RULE 169Theore
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8.6.L’HÔPITAL’S RULE 171and f
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8.7. EXERCISES 1738.6.1 Interest Co
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8.8. MULTIPLICATION OF POWER SERIES
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8.9. EXERCISES 177You can go on cal
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8.10. THE FUNDAMENTAL THEOREM OF AL
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8.11. SOME OTHER THEOREMS 181By 8.3
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8.11. SOME OTHER THEOREMS 183Proof:
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8.11. SOME OTHER THEOREMS 185Theore
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Chapter 9The Riemann And RiemannSti
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9.1. THE DARBOUX STIELTJES INTEGRAL
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9.2. EXERCISES 191Here is an intere
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9.2. EXERCISES 193Show x → V [a,x
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9.2. EXERCISES 195Proof:Let ε > 0
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9.2. EXERCISES 197Thus, since ε is
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9.2. EXERCISES 199Proof:This follow
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9.2. EXERCISES 201I need to verifyU
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9.3. EXERCISES 2039.3 Exercises1. L
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9.4. THE RIEMANN STIELTJES INTEGRAL
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9.5. EXERCISES 2219.5 Exercises1. L
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9.5. EXERCISES 22315. Let there be
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9.5. EXERCISES 225where M ≥ max {
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9.5. EXERCISES 22725. Suppose f is
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9.5. EXERCISES 229which is bounded
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9.5. EXERCISES 231First consider(
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Chapter 10Fourier Series10.1 The Co
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10.2. DEFINITION AND BASIC PROPERTI
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10.3. THE RIEMANN LEBESGUE LEMMA 23
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10.3. THE RIEMANN LEBESGUE LEMMA 23
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10.4. DINI’S CRITERION FOR CONVER
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10.4. DINI’S CRITERION FOR CONVER
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10.5. INTEGRATING AND DIFFERENTIATI
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10.5. INTEGRATING AND DIFFERENTIATI
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10.6. WAYS OF APPROXIMATING FUNCTIO
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10.7. EXERCISES 2556. Let f (x) = x
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10.7. EXERCISES 25714. Suppose |u k
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10.7. EXERCISES 259Now show thatNex
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Chapter 11The Generalized RiemannIn
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11.1. DEFINITIONS AND BASIC PROPERT
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11.2. INTEGRALS OF DERIVATIVES 271b
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11.3. EXERCISES 273The following ex
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Appendix AConstruction Of RealNumbe
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277Proof: ⇐= is obvious. Suppose
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279Next consider the first term. Fo
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Bibliography[1] Apostol, T. M., Cal
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Index∩, 14∪, 14δ fine, 261n th
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INDEX 285future value of an annuity
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INDEX 287Stirling’s formula, 220,
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