# 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 *y*_{n} 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 *a*_{n} 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 *y*_{n} - a and sum of the absolute values of *x*_{k} - 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 *x*_{k} 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