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

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Unformatted text preview: Homework 5 - Math 139 Due Oct 6th Instructor Office Office hours Web page Mauro Maggioni 293 Physics Bldg. Monday 1:30pm-3:30pm. 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 {an } and {bn } be Cauchy sequences. Let {an } ∼ {bn } mean that an − bn → 0 1 . Prove that ∼ is an equivalence relation: {an } ∼ {an }; if {an } ∼ {bn } then {bn } ∼ {an }; if {an } ∼ {bn } and {bn } ∼ {cn }, then {an } ∼ {cn }. 2. Prove that the sum and product of Cauchy sequences is Cauchy. 3. Let [an ] denote the equivalence class of the Cauchy sequence {an }. Given Cauchy sequences {an } and {bn }, define the sum and product of the equivalence classes containing them by [an ] + [bn ] := [an + bn ] [an ] · [bn ] := [an bn ] Prove that these rules are well-defined by showing that if {an } ∼ {a′n } and {bn } ∼ {b′n }, then {an + bn } ∼ {a′n + b′n } and {an bn } ∼ {a′n b′n } 4. If C denotes the set of equivalence classes of Cauchy sequences, then with the sum and product operations in 3. C is in fact a field. Don’t try to prove this, but identify 0 and 1 in C and verify that [an ] + 0 = [an ] and [an ] · 1 = [an ] for all Cauchy sequences {an }.(Keep in mind that your choice of 0 (or 1) in your answer will be an equivalence class of Cauchy sequences. This class may be identified by any Cauchy sequence in it.) 1 Note that this is a different notion of equivalence than the one I gave and used in class. It is in fact an equivalent notion, in the sense that the equivalence classes are the same, but you do not need to show that (albeit that could be an exercise!). Use the equivalence relation defined here to do this problem. ...
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This note was uploaded on 01/16/2012 for the course MATH 139 taught by Professor Pardon,w during the Fall '08 term at Duke.

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