Homework Assignments

Unless otherwise stated, all problems are taken from Lay's Analysis (5th edition). Problem sets should be stapled together, with your name and collaborators clearly identified, and turned in to your instructor at your Thursday recitation section. Because we will post selected solutions on this page soon after the due date, no late homework will be accepted. See the course information sheet for information on how homework is graded.

• Assignment 1, due Thursday, January 25th
• 1.1: 4, 7ab (meaning, here and elsewhere, parts a and b only), 8a, 9ace, 12.
1.2: 4, 5, 7, 8, 10ab, 12abcf, 14.
1.3: 3, 4.
Note: The problems from Section 1.3 above are officially moved to be part of homework 2, but we give the solutions in the solutions set for Homework 1 since some students might be working ahead and want to see the answers now.

• Assignment 2, due Thursday, February 1
• 1.4: 3, 7, 8, 15, 18, 21, 28
2.1: 4(acfg), 7, 10, 15, 16, 20, 22
2.2: 11(a-f), 12(a-d), 14, 17, 30, 31(abde) .
The problems from 2.1 and 2.2 above are pushed to homework 3. We provide solutions now so that students who worked them can see some example solutions.

• Assignment 3, due Thursday, February 5
• 2.2 13,21,25,26,28
2.3:3 a,b,d; 5b;6c;9;10;19
Note: We will definitely push the problems from section 2.3 to next week's homework since we spent extra time reviewing in class to day for the midterm. So, this week's homework (and the subject of next Tuesday's quiz, is tentatively set as the problems from 2.1 and 2.2 above (note the problems in 2.2 are listed in two different places above!). UPDATE (February 7, 2018): We were able to discuss a lot of great questions in review during class today and so any questions from the homework about ``equivalence relations" will be postponed since we did not talk about those questions in lecture. The homework due Thursday (and the subject of the next quiz) is therefore updated as follows: Section 2.1: 4(acfg), 7, 10, 15, 16, 20, 22; section 2.2: 11 a-f; 12 a-d; 31 a,b,d,e; We will still post all solutions in case anyone worked ahead and wants to see the solutions!

• Assignment 4, due Thursday, February 15th
• 2.2 13,21,25,26,28
2.3:3 a,b,d; 5b;6c;9;10;19

2.4: 1;3a,c,d,e; 4;5;9a,b (note: you can use here, without proving it, the fact that any polynomial with real coefficients has a finite number of roots.);11;15;17;21
3.1:3;6;9;14;17;20
Note: the problems above from section 3.1 are postponed until next week's homework and they are not due on Thursday, February 15th!
Note also: be careful when using ordinal numbers that you are comfortable with exactly what they represent. For example - problem 15 on the above homework is not straightforward at all to do - in fact as far as I know, you have to use the Theorem mentioned in Problem 13 in the text (the Schroeder-Bernstein theorem.)

• Assignment 5, due Thursday, February 23rd
• 3.2: 2;3a,b,e,i;4;6;7;10;12
3.3:2;3a,c,e,i;4a,c,e,i;6;10;11;14.
IMPORTANT: WE NEVER DID CATCH UP TO THE ABOVE MATERIAL IN TIME TO MAKE THAT DUE DATE REASONABLE...SO THIS WAS NEVER HANDED IN ON THURSDAY, FEBRUARY 23RD. SEE NEXT HOMEWORK - WE ARE JUST DUPLICATING THE ABOVE AND MAKING IT DUE ON THURSDAY, MARCH 1RST.

• Assignment 5, due Thursday, March 1rst
• 3.2: 2;3a,b,e,i;4;6;7;10;12
3.3:2;3a,c,e,i;4a,c,e,i;6;10;11;14.
Students are very, very, very strongly encouraged to tackle the first half of this homework before the end of the weekend - you have had a lecture on that material and you are all ready to start solving the problems. It takes a while to get the hang of these problems so please start early! (Note: We do expect the students to write up very carefully the homework questions for section 3.1. However in view of the fact that some students were working ahead and did these problems before they were postponed, we already posted the solutions to the 3.1 problems (see above - solutionks to homework 4). Students should write solutions very carefully for these problems as they will show up on quizzes and exams. Those problems will not be graded as part of the homework though, because the solutions are posted above. (We strongly suggest students work the problems before consulting the solutions.)

• Assignment 6, due Thursday, March 8th
• 3.4 1a,b,c,d,h,i;3a,b,c,e;4a,b,c,e;5a,b,c;6a,b,c;7a,c,d;10;15

• Assignment 7, due Thursday, March 22th
• 3.5: 2a,b,c,d,e;3a,d,e;4;5;6,7,8,9,12
4.1:2 a,b,d; 4;6a;7b,c;8b,d;12.
4.2: 3a;5a (More will be assigned from this section on next homework.)
NOTE: Sections 4.1 and 4.2 pushed to next week. Quiz on Tuesday, March 27 will be on Section 3.5. Note also that Section 3.6 is not part of the course.

• Assignment 8, due Thursday, March 29th
• 4.1:2 a,b,d; 4;6a;7b,c;8b,d;12.
4.2: 3a;5a,c,i,j; 6;9; 14;15;16;17.
4.3:3a,d;4;8;9.
Note: problems from 4.3 are postponed to next week, so only the first two sets of problems above are due tomorrow (Thursday).
. Note this file is missing a solution to problem 5j. To do this, first remark that we discussed the following limit in class that is useful here: The limit of (1 + 1/n)^n as n goes to infinity is the number e. (Between 2 and 3). We will prove carefully this limit later in our lectures - but we use that limit here. Indeed, apply the ratio test to the sequence in 5j and you will see using the fact we just gave that the limit is 1/e which is less than 1. So the sequence converges to zero by the ratio test. (There is some algebra needed to show that s_{n+1} / s_n = 1 / (1+1 / n)^n but you can do that yourself or ask a TA or the instructor to show that algebra to you. Once you have this equality you take the limit as n goes to infinity and use the limit laws and the result mentioned above about the limit of the denominator here.)

• Assignment 9, due Thursday, April 5th
• 4.3:3a,d;4;8;9.
4.4: 3a,b,d;7;13;16;17, 18
(there will be no additional sections coverd on this homework!)

• Assignment 10, due Thursday, April 12th
• 8.1: 3;4c,d;5b,d,f;7;11
8.2: 1a,b;2a,b,c;3b,d,f;5b,d,f
8.3: 3a,c,e,g,i;5a;7
Attention: (4-11-18): The problems from section 8.3 here will be due next week instead!!

• Assignment 11, due Thursday, April 19th
• 8.3: 3a,c,e,g,i;5a;7
Note: No quiz on Tuesday, April 17th because of the midterm on Thursday, April 19. Midterm is on sections: 3.4, 3.5, 4.1, 4.2, 4.3, 4.4, 8.1, 8.2.

• Assignment 12, due Thursday, April 26th
• 5.1:3a,b,c,d;4;6b;7b,c;13;18

5.2: 3;5;10;14;16;17

5.3: 3a,b;4;7;8;16.
. Please note: Solution to problem 18 from section 5.1 is missing here! Here is a solution to that problem!. Also: Problem 7 in section 5.3 is interesting and really fun. It's a beautiful application of the Intermediate Value Theorem. Warning: the solution provided here has some typos! They are easy to fix, so I leave you to find them and correct them. (Hint: they have to do with the direction of certain inequalities. That is a lot of the inequalities in the solution are written incorrectly and should be corrected by reversing the inequality. Again, we just want to alert you to this and leave you to find and correct these mistakes! (For example: the very first "less than or equal to" that appears in the solution should actually be a "greater than or equal to" - i.e. the quantity f(a) - a is actually greater than or equal to zero. Do you see why this is true? It's because f(a) is in the interval [a,b] so it is at least as large as a.)

• Assignment 13, Due Thursday, May 3.
• 6.1: 4a,e;6

6.2: 5b,c,h;8b;12

6.3: 3a,b,f;4e

6.4: 3, 4, 5
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