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### Homework_5

Course: MATH 139, Fall 2010
School: Duke
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Word Count: 298

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5 Homework - Math 139 Due Oct 6th Instructor Oce Oce hours Web page Mauro Maggioni 293 Physics Bldg. Monday 1:30pm-3:30pm. www.math.duke.edu/ mauro/teaching.html Reading: from Reeds 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...

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5 Homework - Math 139 Due Oct 6th Instructor Oce Oce hours Web page Mauro Maggioni 293 Physics Bldg. Monday 1:30pm-3:30pm. www.math.duke.edu/ mauro/teaching.html Reading: from Reeds 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 }, dene 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-dened by that showing if {an } {an } and {bn } {bn }, then {an + bn } {an + bn } and {an bn } {an bn } 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 eld. Dont 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 identied by any Cauchy sequence in it.) 1 Note that this is a dierent 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 dened here to do this problem.
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Duke - MATH - 139
Homework 6 - Math 139Due Oct 13thInstructorOceOce hoursWeb pageMauro Maggioni293 Physics Bldg.Monday 1:30pm-3:30pm.www.math.duke.edu/ mauro/teaching.htmlReading: from Reeds textbook: Section 3.3Problems: 3.1: #3,7,8,10I suggest you pick and do
Duke - MATH - 139
Homework 7 - Math 139Due Oct 20thInstructorOceOce hoursWeb pageMauro Maggioni293 Physics Bldg.Monday 1:30pm-3:30pm.www.math.duke.edu/ mauro/teaching.htmlReading: from Reeds textbook: Section 3.4Problems:3.2: #7, 8, 9, 113.3: #1, 2, 3
Duke - MATH - 139
Homework 8 - Math 139Due Oct 27thInstructorOceOce hoursWeb pageMauro Maggioni293 Physics Bldg.Monday 1:30pm-3:30pm.www.math.duke.edu/ mauro/teaching.htmlReading: from Reeds textbook: Section 4.1,4.2Problems:3.3: #8, 13, 154.1: #2,3,5,9,10,12
Duke - MATH - 139
Homework 9 - Math 139Due Nov. 3rdInstructorOceOce hoursWeb pageMauro Maggioni293 Physics Bldg.Monday 1:30pm-3:30pm.www.math.duke.edu/ mauro/teaching.htmlReading: from Reeds textbook: Section 4.3,4.5,4.6Problems:4.2: #2, 3, 4, 5, 6, 124.3: #7,
Duke - MATH - 139
Homework 10 - Math 139Due Nov. 10thInstructorOceOce hoursWeb pageMauro Maggioni293 Physics Bldg.Monday 1:30pm-3:30pm.www.math.duke.edu/ mauro/teaching.htmlReading: from Reeds textbook: Section 5.1,5.2Problems:4.5: #6,8 (for (b) use #6)4.6: #2
Duke - MATH - 139
Homework 11 - Math 139Due Nov. 17thInstructorOceOce hoursWeb pageMauro Maggioni293 Physics Bldg.Monday 1:30pm-3:30pm.www.math.duke.edu/ mauro/teaching.htmlReading: from Reeds textbook: Section 5.3,5.6Problems:5.1: #1,4,115.2: #9,15Additional
Duke - MATH - 139
Homework 12 - Math 139Due Thu Dec. 2ndInstructorOceOce hoursWeb pageMauro Maggioni293 Physics Bldg.Monday 1:30pm-3:30pm.www.math.duke.edu/ mauro/teaching.htmlReading: from Reeds textbook: Section 5.7, 6.1 (review), 6.2. I would suggest alsoread
Duke - MATH - 139
Homework 13 - Math 139Due Wed Dec. 8thInstructorOceOce hoursWeb pageMauro Maggioni293 Physics Bldg.Monday 1:30pm-3:30pm.www.math.duke.edu/ mauro/teaching.htmlReading: from Reeds textbook: Section 6.3, 6.4Problems:6.2: #1,5,8,96.3: #3,6
Duke - MATH - 139
Oce Problem 1 - Math 139Due Oct 13thInstructorOceOce hoursWeb pageMauro Maggioni293 Physics Bldg.Monday 1:30pm-3:30pm.www.math.duke.edu/ mauro/teaching.htmlOce consultation: Oct. 7th, 8th, 1pm-3pm, usual oce hours on Oct. 11th. Possiblyother ti
Duke - MATH - 139
Oce Problem 2 - Math 139Due midnight Dec. 10thInstructorOceWeb pageMauro Maggioni293 Physics Bldg.www.math.duke.edu/ mauro/teaching.htmlIn this Oce Problem you will be constructing the Riemann integral for functionsf : Rn R with n 1.(i) You will
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