Fall 1999 Syllabus

REAL ANALYSIS. In terms of preparation for graduate-level mathematics study, this senior-level mathematics course is one of the two most important courses offered in the Mathematics Department since questions on the Mathematics Graduate Record Examination are drawn more heavily from Real Analysis than from most other areas. In this course basic algebraic and topological properties of the real number system will be established in a precise way and then applied to the rigorous study of such concepts as limit, continuity, differentiation, integration, and infinite series.
Prerequisites: MAT 315 and 330, or consent of instructor.

• To more fully understand the underlying reasons for the major concepts and techniques of calculus.

• To develop skills in proving mathematical theorems in analysis.

• To practice communicating mathematical ideas to others in a precise way.

• To become a more independent learner, logical thinker, and stronger mathematician.

The members of this class constitute a learning community. Learning in such a community best takes place in an atmosphere in which instructor and the students treat everyone with mutual respect. Students need not always raise their hands in order to ask questions or to make comments, but they should not interrupt the instructor or fellow students in doing so. Students typically find the atmosphere set by the instructor to be a sometimes playful and nearly always relaxed one, but students will still need to work hard and consistently both in and out of class in order to do well. If at anytime you have thoughts, comments, or suggestions about how the class atmosphere could be improved or made into one which is more supportive of your learning, please come by or drop me a note about it. I welcome such suggestions.

The main text Fundamental Ideas of Analysis is by Michael Reed. We will cover chapters 1-4 and hope to cover chapters 5-6 as well:

Chapter 1, Preliminaries

This chapter serves as a review of some of the material encountered in MAT 315. We will review the formal properties of the real numbers and sets, establish notation for sets and functions, and remind ourselves about some of the important techniques needed for proving theorems.

Chapter 2, Sequences

In our calculus classes we dealt with the concept of the limit on a rather intuitive level and then used these ideas to intuitively understand some of the concepts of calculus. In this chapter we will more rigorously explore concepts of analysis through the formal study of limits and sequences.

Chapter 3, The Riemann Integral

Like the limit, in our calculus classes we dealt with the concept of the continuity on a rather intuitive level. In this chapter we will rigorously explore the Riemann Integral after a formal study of continuity. We will also explore error estimation in numerical methods of integration.

Chapter 4, Differentiation

In this chapter we will formally develop the notion of the derivative and then we will carefully develop the fundamental theorem of calculus.

Chapter 5, Sequences of Functions

Unlike the previous chapters, this one is not a more formal study of ideas we have already seen in calculus. In this chapter we will study the new idea of the pointwise and the uniform convergence of sequences of functions and sets of functions. In addition, the new topological idea of the metric space is introduced.

Chapter 6, Series of Functions

In this chapter we explore more new ideas. We will explore convergence of series of functions through use of new tools such as the limit superior and limit inferior.

Our course home page is located at http://www.berea.edu/Math/Faculty/Jan/MAT434/.
Use this page as a resource to find this syllabus and other course-related information.
A couple of other great on-line resources are:

• Interactive Real Analysis: http://www.shu.edu/projects/reals/reals.html
  
• Analysis WebNotes: http://www.math.unl.edu/~webnotes/contents/contents.htm

For the benefit of the students in the class, all course grade computations are continually updated by the instructor, so students may check frequently on their in-progress course grade during the term.

Tests and short quizzes will be given in this course. Approximately one announced quiz will be given each week in which there is no test. In general, the announced quizzes will consist of questions on the assigned text readings or homework-like problems.

The three test dates are not pre-scheduled as the instructor believes it is very important that students have input into when the tests are held. However, the tests will fall approximately in the following weeks:

• Week of October 4
• Week of November 1 or 8
• Week of December 6

The comprehensive final exam will likely be a take home final, but if it is an in-class final then it will be at the regularly scheduled time of 3:00 PM Monday, December 13.

Class lectures and discussions are considered to be vital to success in this course. It is the hope of the instructor that class sessions are both informative and useful, therefore attendance is expected at each class session unless a specific exception is made. This policy will be enforced in several ways. Quizzes may be announced or occasionally "popped," and because the lowest quiz grade will be dropped, under nearly all circumstances, make-up quizzes will not be given. Likewise, make-up tests will under almost no circumstances be given, so missed tests will therefore count as the student's dropped 100 points. Absences from class are noted, and repeated absences will adversely affect the student's grade. The final grade may be lowered by one third of a letter grade for each absence after the fourth. Thus, it is the responsibility of the student to speak to the instructor about each absence from class. This should be done as soon as possible, and if at all possible before the absence occurs. Students who miss class are held responsible for all of the material covered, assigned, and collected during their absence.

All written work should be neat, organized, and should show sufficiently many steps to demonstrate a clear understanding of the techniques used. Homework is due at the beginning of class on the announced date due. If a student must miss class due to either a sickness or a planned absence, homework is still expected to be submitted on time. Assignments may be requested in advance. Homework assignments may be turned in before class or at the instructor's office, but should NOT be sent through the CPO. Assignments not meeting the above standards may receive reduced credit.

Learning to collaborate effectively is not only welcomed but is strongly encouraged. Some homework assignments will be specifically designed for teamwork, others for individual work, but on most homework you can choose to work alone or in a team. However, every student must hand in their own individual handwritten completed assignment. On any assignment where some work was completed collaboratively, the assignments should be handed in with each author acknowledging all of the help received for each problem. This includes significant help received from the instructor or by the Math Lab Consultants. Remember that thoughtful practice, not (even mindful) copying, is ultimately the best way to learn.
Warning: Please be careful to conform to these standards for teamwork, since they are designed to encourage good learning practices. (Furthermore, copying another's work or otherwise failing to adhere to these standards may even result in a charge of academic dishonesty.)

There will not be a teaching assistant for this course. Students are strongly encouraged to work together and to make use of the help available in the instructor's office hours. Best results are obtained trying to solve problems alone or in a group before asking for help, so students should be prepared to show what they have already tried. Topics in this course build throughout the course, so students should be sure to do their best to keep up with the class, so as to not get behind and possibly forever lost. Remember, no question to which one does not know the answer is ever "dumb" unless it goes unanswered because it remained unasked.