Bartle Introduction To Real Analysis Homework Solutions

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Homework, Exams and Grading:There will be two in-class midterm exams and one final exam. You are welcome and encouraged to discuss homework with other students, but you must write your solutions individually. The course grade will be determined as follows: Homework (25%). Will be assigned every class; due every Friday before class. We will spend a half of each Friday class discussing the solutions. One homework with the lowest score will be dropped. Scroll down to see the homework. Midterm Exam 1 (20%), Friday, October 16, in class. Books and calculators are allowed. Midterm Exam 2 (20%), Friday, November 20, in class. Books and calculators are allowed. Final Exam (35%), Friday, December 18, 1:30 - 3:30 pm, in USB 2260. Books and calculators are allowed.

Wednesday, September 9 Section 1.2: Principle of Mathematical Induction (without proof); examples. Study preliminaries: Section 1.1 (sets) and Appendix A (logic). Study Principle of Mathematical Induction with an arbitrary base instead of 1 (Statement 1.2.3). Do problems 1-5; they are due Friday, September 18.Friday, September 11Section 1.3: Countable sets and its properties; countability of the set of rational numbers. Note: please correct the statement of the criterion of countability given in class: (i) should be "S is a countable set" instead of "countably infinite". Do problems 6-7; they are due Friday, September 18. Problems 4,5,6,7 will be graded.Monday, September 14 Section 1.3: Countable union of countable sets; uncountability of the power set of N; uncountability of R by Cantor's diagonalization method. Home reading: Sections 2.1-2.2 (simple properties of real numbers). You do not need to memorize all those axioms and properties, but you should be very comfortable with the material in those sections. Pay attention to arithmetic-geometric mean inequality, Bernoulli inequality, and especially triangle inequality and its consequences. Do problems 8-9; they are due Friday, September 25.Wednesday, September 16 Section 2.3: supremum and infimum. Study neighborhoods at home (Def. 2.2.7 and discussion below).Friday, September 18Section 2.4.1: properties of supremum and infimum. Study 2.4.3 (Archimedian Property). Do problems 10-16; they are due Friday, September 25.Monday, September 21Applications of Completeness Axiom: existence of the square root (Theorem 2.4.7); nested intervals (Theorem 2.5.2). Study density of Q (Theorem 2.4.8). Modify the argument given in class to show that there exists a real solution to the equation x^2 = a for every a > 0.Do problems 17-19; they are due Friday, September 25. Problems 9c, 12, 15, 17a, 19. will be graded. Problem 9c = 5 points, all others = 10 points each. Wednesday, September 23Sequences and their limits (Section 3.1).Do problems 20-21; they are due Friday, October 2.Friday, September 25Three remarkable limits (Examples 3.11 b, c, d).Do problems 22-23; they are due Friday, October 2.Monday, September 28Limit theorems (Theorems 3.2.2, 3.2.3).Do problems 24-26; they are due Friday, October 2. Problems 20b, 20c, 21, 22, 26 will be graded; each problem = 10 points.Wednesday, September 30Squeeze theorem (what we did in class roughly corresponds to Theorems 3.2.4, 3.2.5, 3.2.7). Examples. Please study Theorems 3.2.10 and 3.2.11, which we have not covered in class.Do problem 27; it is due Friday, October 9.Friday, October 2Monotone Convergence Theorem (Section 3.3). Application to convergence of recursively defined sequences.Study Sections 3.3.4, 3.3.5.Do problems 28-31; they are due Friday, October 9.Monday, October 5Euler's number e as the limit of the sequence (1+1/n)^n. (Section 3.3.6.) Subsequences (Section 3.4).Do problem 32; it is due Friday, October 9. Problems 27 a, c, d; 29; 31, 32 will be graded. Point values: 21 a, c, d = 5 points each; 29, 31, 32 = 10 points each.Total = 45 points.Wednesday, October 7Bolzano-Weierstrass Theorem (3.4.8). Limit superior and limit inferior.Do problems 33-34; they are due Friday, October 16.Friday, October 9Review for Midterm.Do problem 35; it is due Friday, October 16. Problems 33, 34, 35 will be graded; each problem = 10 points.Here are some practice exams. This one is most representative; the actual exam will be just a bit harder. This exam is too easy but isworth practicing; here are solutions. In this exam, try problems 1 and 7a.Additionally, it is useful to study the worked out examples in the textbook, from each section we covered. There are also problems at the end of each section; they arranged in the ascending level of difficulty. Our exam will be mid-level difficulty, so problems in the middle are most representative. Monday, October 12The Cauchy Criterion. Contractive sequences (Section 3.5).Wednesday, October 14Infinite limits (Section 3.6). Series. Example: geometric and harmonic series (Section 3.7).Do problems 36-41; they are due Friday, October 23.Friday, October 16: Midterm Exam 1. Solutions.Do problems 42-43; they are due Friday, October 23. Problems 36, 37, 39, 40, 42 will be graded. Each problem = 10 points.Wednesday, October 21Comparison tests for series (Section 3.7)Do problems 44-47; they are due Monday, November 2. (This is not a typo.)Friday, October 23Limit of a function (Section 4.1)Do problems 48-49; they are due Monday, November 2. Problems 45, 47, 48, 49 will be graded. All problems = 10 points each. Monday, October 26Limit theorems (Section 4.2)Wednesday, October 28Limit theorems continued (Section 4.2). Some classical limits.Friday, October 30Extensions of the concept of limit (Section 4.3). Continuous functions (Section 5.1).Do problems 50-54; they are due Friday, November 6.Monday, November 2Lipschitz functions. Combinations of continuous functions (Section 5.2). Boundedness Theorem (5.3.2).Do problems 55-56; they are due Friday, November 6. Problems 50 b,d, 52, 53, 56 will be graded. All problems = 10 points each, total = 40 points.Wednesday, November 4Maximum-minium Theorem (5.3.3), Intermediate Value Theorem (5.3.7), preservation of intervals (5.3.9).Do problems 57-59; they are due Friday, November 13.Friday, November 6Uniform Continuity (5.4.3), Continuous Inverse Theorem (5.6.5).Do problems 60-61; they are due Friday, November 13.Monday, November 9Differentiation: definition and basic properties of the derivative (6.1.1-6.1.5). Linearization of functions.Do problems 62-63; they are due Friday, November 13. Problems 57, 58, 59, 62, 63 will be graded; each = 10 points.Wednesday, November 11Landau notation (not in textbook, see e.g. this Wikipedia article). The chain rule. Derivatives of trigonometric functions.Do problems 64-65; they are due Friday, November 20.Friday, November 13Derivative of inverse functions (6.1.8). Local extrema (6.2.1). Rolle's Theorem (6.2.3) and Mean Value Theorem (6.2.4).Do problems 66-67; they are due Friday, November 20. Problems 64 (a,c), 65, 66, 67 will be graded; each = 10 points. Monday, November 16Some consequences of the Mean Value Theorem (6.2.5 - 6.2.7). L'Hospital's Rule (6.3).Wednesday, November 18Taylor's Theorem (6.4.1). Taylor series. Do problems 68-70; they are due Monday, November 30.Friday, November 20: Midterm Exam 2. Solutions. Here are some practice exams. This exam is most representative; here are solutions. In this exam, try problems 2, 3, 4, 6.Additionally, it is useful to study the worked out examples in the textbook, from each section we covered. There are also problems at the end of each section; they arranged in the ascending level of difficulty. Our exam will be mid-level difficulty, so problems in the middle are most representative. Monday, November 23Riemann Integral: definition and basic properties (Section 7.1).Do problems 71-73; they are due Monday, November 30.Wednesday, November 25Sequential and Darboux criteria of integrability. Class notes.Monday, November 30Integrability of continuous and monotone functions, restrictions and combinations (roughly 7.2.7 - 7.2.10). Class notes.Wednesday, December 2: Prof. Mark Rudelson is teaching this class.The plan for the rest of the course is to cover Sections 7.3, 9.1, 9.2 and 9.3.Do problems 74-75; they are due Wednesday, December 9.Friday, December 4: Prof. Jinho Baik is teaching this class.Returning to series. (?)Do problems 76-81; they are due Wednesday, December 9. Problems 76, 77, 80, 81 will be graded. Each problem = 10 points. Monday, December 7: Prof. Alon Nishry is teaching this class.Absolute and conditional convergence (9.1.1-9.1.2). Limit comparison test and Root test for absolute convergence (9.2.1-9.2.3).Wednesday, December 9Ratio Test (9.2.4-9.2.5) and Integral Test (9.2.6). Do problems 82-84; they are due Monday, December 14.Friday, December 11Alternating series (9.3.2). Grouping (9.1.3) and rearrangement (9.1.5) of the seriesMonday, December 14Review for Final Exam.Friday, December 18: Final Exam: 1:30-3:30 pm in USB 2260Here are some practice exams. This exam is most representative; here are solutions. (I have posted it already before the midterm exam, but Problem 1 about series was not accessible back then.)This exam was also posted before, you should be able to solve all problems there. In this exam, try all problems except 6,7,12.And in this exam, try all problems except 7,9,11.Additionally, it is useful to study the worked out examples in the textbook, from each section we covered. There are also problems at the end of each section; they arranged in the ascending level of difficulty. Our exam will be mid-level difficulty, so problems in the middle are most representative.

Analysis, one of the pillars (along with algebra and topology) of modern mathematics, begins with a rigorous development of single-variable calculus. Thus,this course serves the important purpose of teaching you how to rigorouslyprove and apply results in calculus, including results related to the notionsof limit, continuity, derivative, integral, and infinite series. Inherentin all of these notions is the concept of approximation. As we shallsee, a good grasp of this latter concept is essential not only in proving''pure'' results in analysis, but is also crucial in ''applied'' problemsrequiring estimations. In any approximation a key question is ``howdo you estimate the error''? In the first part of this course,we will look at some types of algebraic manipulations that can be usedin error estimation; we will also look at more powerful methods involvingthe mean value theorem for derivatives.Difference between the courses:MAT 320 is more comprehensive and providesa firm grounding for further study. MAT 319 has more of an emphasis ontopics which arise in high-school calculus.Students planning to go on to graduate schoolin mathematics are advised to take MAT 322 and MAT 324 as well.Students wanting to take MAT 322 or MAT 324 (or the seminarsMAT 401 or MAT 402) will need to take MAT 320, not MAT 319. Studentswho want to take these courses after MAT 319 instead will need to do some extrawork, and get permission from the relevant instructor.Prerequisites: C or higher in MAT 200 or permission of instructor;plus one of the following: C or higher in MAT 203, 205, 211, AMS 261, or A- orhigher in MAT 127, 132, 142 or AMS 161. Anyone lacking these prerequisitesrisks deregistration.Instructors: Daryl Geller, 4-100B Math Building,phone 632-8327. E-mail daryl@math.sunysb.edu - Website ~darylOffice hours: Tuesdays and Thursdays from 12:50-2:20. Check for announcements or postings on the website regularly!Michael Movshev, 4-109 Math Building, phone 632-8271.E-mail mmovshev@math.sunysb.edu - Website ~mmovshevOffice hours: to be announced. Teaching Assistants: Jan GuttE-mail jgutt@math.sunysb.eduXiaojie Wang, 2-106 MathE-mail wang@math.sunysb.eduOffice hours: Mondays and Wednesdays, 2-3 p.m. in the Math LearningCenter, and Mondays 1-2 p.m. in 2-106 Math.Text for the first five weeks and for MAT 320:D. Geller, A Bridge to Analysis, available through lulu.com.You can click on this link to order the book:A Bridge to Analysis Please select Ground shipping, as Mail may take too long.lulu.com is a self-publishing website that makes books available verycheaply.Solutions will be distributed as the semester progresses. There are solutions for almost every problem in the book.Text for MAT 319, after the first five weeks:R.G. Bartle and D.R. Sherbert, Introduction to Real Analysis, 3rd edition,available in the university bookstore. You can buy it after the split.Grading System: The first midterm, on the first three chapters of thebook (up to page 117) will be given in class on Thursday, October 7.It will be graded by Monday, October 11, at which time the classes will split.You will be allowed to switch your registration from MAT 319 to MAT 320 at that time.Your results on the test will help you to make an informed decision.There will also be a second midterm. In MAT 320, it will be held on Tuesday, November 16. The final examination will be held on Thursday, December 16, from11:15 a.m. - 1:45 p.m.Students are expected to ensure when they register for the courses thatthey will be available for the final examination, and that they donot have too many final exams on that date.The final course grades will be determined as follows:homework 10%, two midterms 25% each, final exam 40%.The grades of A- and A will be reserved for students who demonstrate asubstantial ability to apply the concepts of the course in new andsomewhat creative ways.Please note that there will be no curve in this course indetermining grades. Incompletes will be granted only if documented circumstances beyond yourcontrol prevent you from completing the course work.Homework:The only way to learn the material is to work problems for yourself. Eachweek, you should attempt to do all of the problems from the sectionswhich are covered in class. We will ask you to hand some problems in.Your homework will be graded meticulously and will give you vitalfeedback on where you are making mistakes.Homework is a means to an end, the ``end'' being for you tolearn the material. We encourage you to work on homework togetherwith friends. In this course, we will never prosecute anyone foracademic dishonesty on any issue relating to homework.If you hand in complete, correct solutions, you will get fullcredit for them, no matter how you obtained them. When you hand in homework in this course, you are not claiming that it is your own work.If someone regularly ``does''the homework by copying from friends or from solution manuals, they areonly cheating themselves, since this is not a way to learn the material.Moreover, they will not receive the benefits of the feedback that our meticulous grading will provide.Never be shy to ask us how to do a homework problem, even if you handed in a copied solution that you do not understand. You will not beprosecuted or condemned for this, and we will be only too glad to helpyou. Approximate Course Schedule:Chapter 1: Aug 31 -- Sep. 14>Chapter 2: Sep. 16 -- Sep. 28Chapter 3: Sep. 30 - Oct. 5First Test (on Chapters 1-2): Oct. 7`After the split, MAT 320 continues:Chapter 4: Oct. 12 - Oct. 28Chapter 5: Nov. 2 - Nov. 11Second Test (on Chapter 4): Nov. 16Chapter 6: Nov. 18 - Dec. 9After the split, MAT 319 continues:To be announced. Americans with Disabilities Act: If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disability Support Services, ECC (Educational Communications Center) Building, room128, (631) 632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential.Arrangements should be made early in the semester(before the first exam) so that your needs can be accommodated. 2b1af7f3a8