Spring 2012

** Lectures: ** MWF 1:00PM-1:50PM at Thack 627

** Office hours: **
M 11:00-12:00 and Th 3:30-4:30 or by appointment, held at Thackeray Hall 610. Attending office hours is greatly encouraged,
in particular at the beginning of semester.

** Recitation: ** Tu 1:00PM-1:50PM at Thack 524,
Instructor: Mark Tronzo

** Objectives: **
This is an advanced course for highly motivated students. The main goal is that the students acquire
proficiency in reading, understanding and writing mathematical proofs in the standard format
in the context of mathematical analysis as well as familiarize themselves with some more advanced
aspects of the topic.
This is an intense writing course and every student will have to complete 2-3 writing projects.
The text has to be typed and hand written texts will not be accepted. The projects will be announced
later.

** Textbook: **
There is no required textbook. The following resources are recommended.

Bartle and Sherbert, Introduction to Real Analysis,

Third Edition, Wiley.

You can refer to

Principles of Mathematical Analysis, Third Edition.

Walter Rudin

for some occasional more advanced material.

For a more detailed discussion of the topics and in particular for a thorough review of the
first part of the course on logical semantics and set theory, you can refer to
Professor Hajlasz's lecture notes:
Introduction to Analysis. (based on MATH0450, Spring 2010 notes, courtesy of Piotr Hajlasz).

** Grading: ** Your final grade depends
on your performance on the final exam as well as on your total grade
calculated as follows:

Homework (10%), Written projects (10%), Class midterms (40%), Final (40%).

** Important notices about homeworks, written projects and exams: **

Clarity and tidiness of presentation, completeness and correctness are necessary
to obtain full grade. In particular untidy presentations of correct proofs will be severely
penalized.

You may work with other students or use library resources, but each student must
write up his or her solutions independently. Copying solutions from other students
will be considered cheating, and handled accordingly.

There will be no make up midterm exams. If you miss the midterm exam for
a *documented* medical reason, your grade on it will be the prorated grade
of your final exam.

** Extra Material:**

A short review of logic.

** Homework:**

Homework 1.

Homework 2.

Homework 3.

Homework 4.

Homework 5.

Homework 6.

Homework 7.

Homework 8.

Homework 9.

Homework 10.

Homeworks are due on Tuesdays and have to be handed in to the TA.

Practice exercises for cardinality.

Practice exercises for Archimedean property, induction and inequalities.

Practice Midterm 1.

Final Exam Guidelines.

Midterm 1 cut-off: All covered material up to and including the topic on Archimedean property of the real numbers.
That corresponds to the first three chapters of the online notes, but we also covered some extra material such as equivalence and order
relations in class.

Disclaimer: The practice midterm is to give you a general idea of the type and level of the problems of the midterm exam.
The problems in the exam are NOT necessarily very similar or the same as in the practice exam.

** Written Projects: **
It is required that the projects be typed on a computer. Using TeX or LaTeX is preferred, but you can also use Word or other systems.
To encourage using TeX or LaTeX, 1 bonus point for each project
typed in that system will be accorded. Here is a
LaTeX template.

Here is a comprehensive online guide for LaTeX .

** Written Project 1:**

Assuming the premises and basic facts of the axiomatic set theory, write the definitions of the notions of
equivalence relations on a set and
partition of a set. State and write the proof of the fundamental theorem of equivalence relations and provide an illustrating example.
Your paper must be written in such a form that your proof could possibly be included in a hypothetical textbook.
A bonus point will be accorded to the most elegant and original expository examples.

There will be no homework due on Tue. Feb. 14. I will collect your written projects on ** Wed. Feb. 15 ** at the beginning
of class. After grading them I will return them to you for corrections (if there will be such a need) and then I will grade
them again. Only the score from the final grading will count.

** Written Project 2:**

Choose one of the three topics below. Your paper must be written in such a form that it could
possibly be included in a hypothetical textbook. If you need to include pictures, leave the space and draw the pictures by hand.

1. Write a paper in which you state and prove the Cantor-Bernstein theorem.
Include the proofs of the lemmas which were left out in the class.

2. Write a paper in which you construct a complete ordered field using cuts in the set of rational numbers.
Refer to the discussion in the class on Monday Feb. 20th for some guidelines. You should prove all the properties of such a field.

3. Assuming the existence of real numbers, construct an ordered field which includes them but is not complete.
Refer to the discussion in the class on Monday Feb. 20th for some guidelines. You should prove all the properties of such a field
and show that it cannot be complete.

There will be no homework due on Tuesday March 13. I will collect your written projects on ** Wed. Mar. 14 **
at the beginning of class. After grading them I will return them to you for corrections (if there will be such a need)
and then I will grade them again. Only the score from the final grading will count.

** Written Project 3, The Term Paper:**

The term paper should be written in the style of an expository article on a topic of your choice
in Analysis. Here is
a list of suggested topics , but you are not limited to that list. If necessary, I could give you some further personal
advice on the choice of your topic.

Your paper should be interesting and
comprehensible to a reader who does not already know your topic. It must include some technical detail or proofs
and some expository examples and explanations. It is not necessarily required that all minor proofs are presented.
It must be typed/word-processed, and should be approximately eight pages in length, double spaced. All sources
must be cited in the paper. Papers that include passages lifted verbatim from
other sources without appropriate citations will receive a grade of F.

The timetable for the term paper is as follows:

** April 6: ** Submit and get your topic approved by me.

** April 11: ** Submit an abstract of your paper in class.

** April 26: ** Finished paper due at 5 pm.

** Midterm exams: ** They will be in class. The dates are ** Friday 2 March ** and ** Friday 30 March**.

** Final Exam: ** The final exam will be held on ** Monday April 23rd, 4:00-5:50 pm.**
The exam room will be Cathedral of Learning room 244B.