Mathematical Analysis I, 2004a

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6 Chapter 1. Set Theory (“each n ∈ N is exceeded by some m ∈ N”) is true, but (∃ m ∈ N) (∀ n ∈ N) m>n is false. However, two consecutive universal quantifiers (or two consecutive existential ones) may be interchanged. We briefly write “(∀ x, y ∈ A)” for “(∀ x ∈ A) (∀ y ∈ A),” and “(∃ x, y ∈ A)” for “(∃ x ∈ A) (∃ y ∈ A),” etc. We conclude with an important remark. The universal quantifier in a formula (∀ x ∈ A) P (x) does not imply the existence of an x for which P (x) is true. It is only meant to imply that there is no x in A for which P (x) fails. The latter is true even if A = ∅; we then say that “(∀ x ∈ A) P (x)” is vacuously true. For example, the formula ∅⊆B, i.e., is always true (vacuously). (∀ x ∈∅) x ∈ B, Problems in Set Theory 1. Prove Theorem 1 (show that x is in the left-hand set iff it is in the right-hand set). For example, for (d), x ∈ (A ∪ B) ∩ C ⇐⇒ [x ∈ (A ∪ B) andx ∈ C] ⇐⇒ [(x ∈ A or x ∈ B), and x ∈ C] ⇐⇒ [(x ∈ A, x ∈ C) or(x ∈ B, x ∈ C)]. 2. Prove that (i) −(−A) =A; (ii) A ⊆ B iff −B ⊆−A. 3. Prove that A − B = A ∩ (−B) =(−B) − (−A) =−[(−A) ∪ B]. Also, give three expressions for A∩B and A∪B, in terms of complements. 4. Prove the second duality law (Theorem 2(ii)).
§§1–3. Sets and Operations on Sets. Quantifiers 7 5. Describe geometrically the following sets on the real line: (i) {x | x
Page 1: The Zakon Series on Mathematical An
Page 4 and 5: Copyright Notice Mathematical Analy
Page 6 and 7: vi Contents 7. Intervals in E n ...
Page 9 and 10: Preface This text is an outgrowth o
Page 11: About the Author Elias Zakon was bo
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56 Chapter 2. Real Numbers. Fields
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Chapter 3 Vector Spaces. Metric Spa
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§§1-3. The Euclidean n-Space, E n
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§§4-6. Lines and Planes in E n 73
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§§4-6. Lines and Planes in E n 75
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§7. Intervals in E n 77 2. In part
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§7. Intervals in E n 79 Corollary
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§8. Complex Numbers 81 We make som
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§8. Complex Numbers 83 where x and
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§8. Complex Numbers 85 Use it to f
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∗ §9. Vector Spaces. The Space C
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∗ §9. Vector Spaces. The Space C
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∗ §10. Normed Linear Spaces 91 E
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∗ §10. Normed Linear Spaces 93 P
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§11. Metric Spaces 95 §11. Metric
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§11. Metric Spaces 97 For a fixed
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§11. Metric Spaces 99 where ¯x =
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§11. Metric Spaces 101 Reverse all
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§12. Open and Closed Sets. Neighbo
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§13. Bounded Sets. Diameters 109 D
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§13. Bounded Sets. Diameters 111 s
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§13. Bounded Sets. Diameters 113 4
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§14. Cluster Points. Convergent Se
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§15. Operations on Convergent Sequ
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§16. More on Cluster Points and Cl
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§17. Cauchy Sequences. Completenes
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150 Chapter 4. Function Limits and
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252 Chapter 5. Differentiation and
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Index Abel’s convergence test, 24
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Index 343 arcwise-, 211 curves as,
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Index 345 as a normed linear space,
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Index 347 bounded, 111 continuity o
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Index 349 limits in one variable, 1
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Index 351 Ordered pair, 1 inverse,
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Index 353 nondecreasing sequences o
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Index 355 Euclidean spaces, 87 norm
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Mathematical Analysis I, 2004a

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