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# Homework_5 - Homework 5 Math 139 Due Oct 6th Instructor Oce...

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Homework 5 - Math 139 Due Oct 6th Instructor Mauro Maggioni Office 293 Physics Bldg. Office hours Monday 1:30pm-3:30pm. Web page www.math.duke.edu/˜mauro/teaching.html Reading : from Reed’s textbook: Sections 6.1,3.2 Problems : § 2.4: #10 § 2.6: #1, 3, 9 § 6.1: #1(a,c), 8 Additional Problem: 1. Let { a n } and { b n } be Cauchy sequences. Let { a n } ∼ { b n } mean that a n - b n 0 1 . Prove that is an equivalence relation: { a n } ∼ { a n } ; if { a n } ∼ { b n } then { b n } ∼ { a n } ; if { a n } ∼ { b n } and { b n } ∼ { c n } , then { a n } ∼ { c n } . 2. Prove that the sum and product of Cauchy sequences is Cauchy. 3. Let [ a n ] denote the equivalence class of the Cauchy sequence { a n } . Given Cauchy sequences { a n } and { b n } , define the sum and product of the equivalence classes containing them by [ a n ] + [ b n ] := [ a n + b n ] [ a n ] · [ b n ] := [ a n b n ] Prove that these rules are well-defined by showing that if { a n } ∼ { a n } and { b n } ∼ { b n } , then { a n + b n } ∼ { a n + b n } and { a n b n } ∼ { a n b n } 4. If C denotes the set of equivalence classes of Cauchy sequences, then with the sum
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