MAT 316, Winter 2018

Homework Assignments

Warmup Problem (due by Friday of Week 1)

Reading: Section 1.1. Then do exercise:
Section 1.2: 2.
Note: Be careful, and make sure you have considered every possibility. This is not for grade.
The solution is now available here.

Week 1 (Jan 8-12) (due by Wednesday of Week 2)

Reading: Sections 1.1-1.3. Then do exercises:
Section 1.2: 1, 4, 7, 9, 11, 12, 13
Section 1.3: 1a, 2.
Notes:

  • What Abbott calls the range of f in problem 1.2.7, I will call the image of f in this class.
  • In 1.2.13b, it is true that induction does not apply, though I don't see why Abbott's problem is an illustration of this point. Regardless, 1.2.13b is a nice problem.
  • Don't forget 1.2.13c, it is hiding on the next page.
  • Solutions have been posted here.

    Week 2 (Jan 16-19) (due by Wednesday of Week 3)

    Reading: Sections 1.3-1.4 and 8.6. Then do exercises:
    Section 1.3: 3, 5, 8ab, 9a, 11
    Section 1.4: 2, 6
    Section 8.6: 5ad, 6.

    Week 3 (Jan 22-26) (due by Wednesday of week 4)

    Reading: Sections 1.4 (for the NIP), 1.5, 1.6 (but not power sets), 2.2. Then do exercises:
    Section 1.4: 3, 8
    Section 1.5: 1, 4c, 6, 9
    Section 2.2: 1, 3, 4.

    Week 4 (Jan 29-Feb 2) (due by Wednesday of week 5)

    Reading: Section 1.2 (for Example 1.2.5), Sections 2.2 and 2.3. Then do exercises:
    Section 2.2: 2, 5, 7
    Section 2.3: 1, 2, 3, 5, 7, 12

  • On Exercise 2.3.3, you can not just use the Order Limit Theorem, as it is not assumed that yn converges.

    Week 5 (Feb 5-9) (due by Wednesday of week 6)

    Reading: Sections 2.3, 2.4 (for monotone convergence theorem and applications), 2.5, 2.6, nothing on series. Then do exercises:
    Section 2.4: 1, 2a
    Section 2.5: 1abc, 2bc
    Section 2.6: 2, 3a

    Week 6 (Feb 12-16) (due by Wednesday of week 7)

    Reading: Section 2.4, 2.7, 2.9 page 83. Then do exercises:
    Section 2.4: 8, 10
    Section 2.5: 3
    Section 2.7: 1c, 2, 4, 7, 9, 10.

  • Somewhere in this homework, exercise 2.3.5 will be useful.
  • For 2.4.10, don't miss the assumption that an is non-negative.
  • For 2.4.10, you should also think about why (1+a)(1+b)(1+c) is greater than (1+a+b+c) when a, b, c are non-negative. (There was a typo in this hint previously, it said less than instead of greater than.)
  • Some comments and solutions have been posted here.

    Week 7 (Feb 19-23) (due by Wednesday of week 8)

    Reading: Review Chapters 1 and 2. Then do the following exercises. Most of them have already been assigned - I want you to rewrite your solutions, with attention paid to well-written final proofs, which take care of every possible case.
    Section 1.3: 6, 9a, 11abc
    Section 1.4: 3
    Section 2.2: 2, 6
    Section 2.3: 2, 5, 11

  • The last problem, 2.3.11, is new and somewhat difficult, but if you can write this up well, you've really got it. Here are some hints. Let a be the limit in question.
  • One can use the triangle inequality to "compare" the absolute value of yn - a and sum of the absolute values of xk - a for each k less than n.
  • One should split this sum into two halves: the half for k not too large, and the half for k large enough. When k is large enough, the difference between xk and a is small. When k is not too large, one can find an upper bound for this difference.

    Week 8 (Feb 26-March 2) (due by Wednesday of week 9)

    Reading: Chapters 4.1, 4.2, 4.3.

  • Note: because I have skipped Ch 3, the phrase limit point appears frequently and is not yet explained. The book is always talking about a limit point of the source of a function. For now, you only need to think about functions defined on the entire set of real numbers. In this case, every real number is a limit point. So when you see limit points, you can just think about real numbers.
    Then do exercises:
    Section 4.2: 2, 4, 5, 6a, 7
    Section 4.3: 1, 8

    Week 9 (Mar 5-9) (due by Wednesday of week 10)

    Reading: Chapters 4.2, 4.3, 4.5 skipping p137 and the bottom of p136, 3.2. Then do exercises:
    Section 4.2: 11
    Section 4.3: 4, 6, 9, 11 (extra credit)
    Section 3.2: 2, 3, 4, 6abcd, 13.

    Ben Elias
    Department of Mathematics
    Fenton Hall, Room 210
    University of Oregon
    Eugene, OR 97403
    Phone: (541) 346-5629
    Fax: (541) 346-0987
    e-mail: bezzzzlizzzzas@uorezzzzgon.edu