# Math 554 Section 001 (MTWTh 10:30--12:45, in LeConte 303B) Analysis I

## Summer I 2012

### 10:30 AM - 12:45 PM, with a 15 minute break

Professor: Peter J. Nyikos

Office: LeConte 406

Phone: 7-5134

Email: nyikos @ math.sc.edu

Usual Office hours:
1:30-3:30 MTWTh

Special Office Hours Monday, May 28 (Memorial Day -- a day which most Americans treat as a holiday): 1:00 - 1:30 pm.

Special Office Hours Wednesday, June 27: 11:00 - 2:00

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 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 very 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.

The midterm was on Thursday, June 14. It covered up through Section 2.4 of the sections covered in class.

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.

Theorems whose proofs you might be asked to give on the final exam are:
1.1.1 (a), 1.4.4, and 1.5.1;
2.2.2 (b), 2.3.2, 2.3.3, 2.4.3, 2.4.5, 2.4.7;
3.1.6, 3.1.7, 3.1.9;
4.1.3;
6.1.3

Concepts you might be asked to define are:
1.2.1, 1.2.3, 1.2.7, 1.4.1, 1.4.3
2.1.6, 2.1.7, 2.1.9, 2.3.6, 2.4.1, 2.4.5, 2.5.1
3.1.1, 3.1.3
4.1.1, 4.1.7, 4.1.11, 4.2.1
5.1.1
upper sum, lower sum, 6.1.1.

You will expected to do problems like some of the following:
1.1, Exercises 1ab, 2c
1.2, Example 1.2.4 (a), Exercises 1, 3, 4, 6
1.3, Exercises 1ab, 2ac, 5a (proof by induction of formula, which will be given)
1.4, Example 1.4.2 (a), Exercises 5 and 14a
2.1, Examples 2.1.8 (a)thru(d) and 2.1.11, Exercises 6, 9abc
2.2, Exercise 11
2.3, Example 2.3.4(a)
2.4, Examples 2.4.2 and 2.4.6, Exercises 3 and 7
2.5, Exercise 1
3.1, Examples 3.1.4(d), 3.1.8; Exercise 4
4.1, Exercises 3ac

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 first homework assignment was due on Wednesday, June 6. A second chance was given on starred items, due Tuesday, June 12. Section 1.1, numbers 5 and 11
Section 1.2, numbers 8(b)* and 9*
Section 1.3, numbers 2(d) and 5(c)*
Section 1.4, numbers 2(b)* 5(d)(j) and 14(a)

The second homework assignment was due on Tuesday, June 19. A second chance will be given on starred items, due Monday, June 25.
Section 1.5, number 7*
Section 2.1, numbers 6d and 8b
Section 2.3, number 4* [compare (d) on page 63]
Section 2.4, numbers 2a*, 3c and 9**
Section 2.5, number 8*, prove "answer" in the back
Section 3.1, finish first three columns [The following were already done in class: [a,b], (a, b) and Q]

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