Mathematics 481-2, 681-2
Real Analysis 
(3 credit hours)


This page: [ Prerequisites | Comments |  Catalog Description | Grade | Presentations | Objectives | Text ]
Teacher:
 Dr. Caldwell, office 429 Holt Humanities, phone 7336. Department office 7360. email: caldwell@utm.edu.
Comments:
 This course and the course text are intentionally difficult. Mathematics 461-2 and 481-2 are the capstone courses for our undergraduate major and are important transition courses for those students continuing on to graduate school. Plan to work hard, and in return, to mature mathematically. Suggestions: (1) Do the homework! (2) Study with a friend (make a new friend if necessary). (3) Use other real analysis books as references. (4) Stay ahead of the class in the text (we will move through it sequentially, skipping some of the optional sections but we will not "hop around"). (5) Come by my office (make appointments as necessary). I want you to succeed and will gladly help you. (6) Review the first chapter (your discrete text should be very helpful). We will begin assuming that you understand most of that chapter. Rather than plan the semester in advance we will move at the classes pace (as determined by me). To help you learn (and to motivate you to keep up) we will try to do a lot of board work (student proofs in class on the board).
Prerequisites:
 Discrete Mathematics (math 241) and Multivariate Calculus (Math 320)
Catalog Description:
 Sets and countability. The real number systems. Sequences, limits, infinite series, metric spaces, continuous functions, uniform continuity, and convergence.  Riemann and Lebesgue integration. Students are required to submit written work and make an oral presentation. 
Grade:
 The course grade will be a weighted average of the homework, tests, final, board-work and, for the 681 students, a research project. 

BRIBES (of the student, the teacher is unbribable):
    1%  for each winning solution to the problem of the month.
    1%  for each solution on the Virginia Tech. Competition
    1%  for attempting each half of the Putnam Competition.
    3%  for each apparently correct solution on the Putnam.
Homework:
 Homework will be assigned and collected daily (at the beginning of the following class period).  No late homework will be accepted.  Much of the homework will be from the text, but we will augment the text wherever necessary.   If you will miss a day, ask ahead of time what the homework will be.
Presentations:
 Early in the semester we will identify student's areas of interest and assign each one topic to present to the class.  This could be material from a section of our text that we do not cover, or some area the student is especially interested in.  The presentation should include homework for the other students. 
Departmental
Objectives:
 The student will: 
  1. Define the real numbers, least upper bounds, and the triangle inequality.
  2. Define functions between sets; equivalent sets; finite, countable and uncountable sets. Recognize convergent, divergent, bounded, Cauchy and monotone sequences. 
  3. Calculate the limit superior, limit inferior, and the limit of a sequence. 
  4. Recognize alternating, convergent, conditionally and absolutely convergent series. 
  5. Apply the ratio, root, limit and limit comparison tests. 
  6. Define metric and metric space. 
  7. Determine if subsets of a metric space are open, closed, connected, bounded, totally bounded and/or compact. 
  8. Determine if a function on a metric space is discontinuous, continuous, or uniformly continuous. 
Text:
 An Introduction to Classical Real Analysis, Stromberg, Wadsworth and Brooks Cole. Adopted in 1991. ISBN 0-534-98012-0.