# Math 554 Section 001 (MTWThF) 10:05--12:05, in LeConte 310) Analysis I

## Session I 2014

Professor: Peter J. Nyikos

Office: LeConte 406

Phone: 7-5134

Email: nyikos @ math.sc.edu

Special Office hours Friday, June 27: 10:00-12:30

The course grade is based 40% on homework and quizzes, 20% on a midterm test, and 40% on the final exam. The final exam is cumulative.

The midterm, some quizzes and the final will include questions about definitions, and you will also be expected to prove a number of things on these two tests, but only things whose proofs may be found in the textbook. Moreover, unless the proof was gone over in class (or an alternative proof provided in class) you will also have the option of proving something that was proven both in class and in the textbook.

The homework, on the other hand, includes problems where you have to come up with your own proofs. Except for easy proofs, you have two chances to come up with the proofs, and I will provide some hints to make it easier for you the second time around.

This course covers parts of Chapters 1 through 6 in Introduction to Real Analysis, 2nd edition, by Manfred Stoll, with emphasis on Chapters 2 and 4. Some material from each of the following sections will be covered:
1.1 through 1.5
Chapter 2 through 2.5
3.1
4.1, 4.2
5.1
6.1
Excerpts from other sections are covered as time permits.

The objectives of this course are to arrive at a a command of the basic concepts of set theory and real analysis up through integration, and the ability to prove statements in set theory and real analysis on the level appropriate to the course. Among the concepts you are expected to master are:

• the set theoretic operations of union, intersection, complementation, and cartesian product;
• functions and their inverses (if any), image and preimage of sets under functions;
• ordered fields, suprema, infima, completeness and Archimedean order of the real line;
• open and closed subsets of the real line, and the interiors of arbitrary subsets;
• boundedness of sets, sequences, and functions;
• limits of sequences, series, and functions including left and right limits, limits at infinity and infinite limits;
• continuity, differentiability and integrability of real-valued functions of a real variable.
Applications and interrelations of these concepts are given all through the course.

The second quiz is on Tuesday, June 10, on Sections 1.2 and 1.4.

Practice problems, not to be handed in:
1.2 number 6 a c
1.4 number 5, starred parts
2.1 numbers 2, 3 (use induction) 6 a c, 8 a c, 9a
2.2 numbers 2b, 6 starred parts, 7 starred parts.
Show 2.2.3 by showing that, for each epsilon > 0, there exists n(epsilon) such that |a_n*M-0| < epsilon for all n > or = n(epsilon), where M is an upper bound for |b_n|. 2.4 numbers 2a, 3ae, 4a, 7ac
2.5 numbers 1acd, 3, 8
3.1 numbers 2, 4, 5, 7a, 8a.
4.1 numbers 1a, 3ac, 8ac, 9, 17acg
5.1 numbers 1bce, 5a, 9b

Hand in Thursday, June 12:
1.4 numbers 5f, 10
2.1 numbers 4f [Hint: use (2) below and class notes], 7d, 8b, 9b
show that if x > 0, y <0 then
(1) y < x+y < x;
(2) |x+y| < max(|x|, |y|) (you may use (1) but nothing after 2.2.2)
Problems 10 and 8b were still eligible up to Thursday, June 19.

Hand in Monday, June 16:
2.2 number 7d, using only results in the book up to that point.
2.3 number 7 (a) (b)
Prove 2.3.7
Prove that if p_n diverges to -(infinity) then every subsequence diverges to -(infinity)

Hand in Monday, June 23:
2.4 4c, 5
2.5 1 b, f
3.1 7b, 8b

Hand in Wednesday, June 25
3.1 8c
4.1 1b, 8b, 13 [Hint: 4.1.3 and Section 2.2]

The final exam in this course is on Saturday, June 28, at the usual class time and place.

#### Extra Credit Problems

Extra credit problems are to be done strictly on your own, except that I am willing to give you advice. You are not to discuss them with anyone else.

There is no due date for extra credit, but once a fully correct solution is handed back, the problem is no longer eligible for extra credit. This is true of any problem crossed out below.

If you can't quite get the solution but have some ideas, hand them in for partial credit. I will keep adding to your score as you improve your work on it.

1. Section 3.1, 7 (b)

2. Section 3.1, 7 (c)

3. Section 3.1, 18 (b)

4. Section 3.1, 18 (e)

5. Section 4.1, 3 (d)

6. Section 4.1, 18

7. Section 5.1, 12b

8. Section 6.1, 18