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

This page: [ Prerequisites | CommentsCatalog Description | Grade | Presentations | Objectives | Text ]


Dr. Caldwell, office 429 Humanities, office phone 7336.  Department office 7360.  E-mail:  Web page:

My main goal in this course is to teach you about this subject that I love.  It may not be obvious, but the way I teach this class (daily homework quizzes, tests, ...) are designed to help you succeed and to make sure all students are treated fairly. Come by my office anytime--too few students take advantage of office hours.  At UT Martin you can talk to and work with your teachers, take full advantage of that!

This course and the course text are intentionally difficult. Mathematics 461-2, 471-2 and 481-2 are "capstone" courses for our undergraduate majors 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 if necessary).  I want you to succeed and will gladly help you.  (6) For the first chapter your Math 314 text should be very helpful. (7) Do the homework! 

To help you learn (and to motivate you to keep up) we may "randomly" do board work (student proofs in class on the board, perhaps in teams of two).  These will be done in a supportive way, using the class as the audience for your proof. If you had me in 314, then you will remember my slogans: "learn to learn," and "write to be read."  They will be emphasized here as well.
Foundations of Mathematics (Math 314) and Multivariate Calculus (Math 320)  (The need for a firm basis in the first of these courses will be obvious the very first day.  The second is basically a maturity requirement: if you can not pass Math 320, you can not as this course.)
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.
Principle of Real Analysis ["Baby Rudin"] 3ed McGraw Hill.  ISBN: 978-0070542358 (0-07-054235-X) hardback or 978-0070856134 (0070856133) paperback.
The course grade will be a weighted average of the homework (19%), tests (53%), final (17%), board-work and project (11%).  For 681-682 students, the above will be 71% of the grade and a research project (paper and presentation) the other 29%.

Bonus points: (if these exam are offered on campus--see Dr. Wagner)
    2%  for each correct solution on the Virginia Tech. Competition
    1%  for attempting each 3 hour half of the Putnam Competition
    5%  for each apparently correct solution on the Putnam Competition
  -1%  for each day less than seven that you come to my office

Homework will be assigned and collected daily at the beginning of the following class period (unless otherwise announced).  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.
Each student must work on a project related to the course content (complete some of the execises or sections of the text we skip, extending one of the concepts...).
Student Learning Objectives:

This course especially addresses the last three of our five student learning outcomes for the major. These are as follows.

The student will: 

  1. apply mathematical concepts and principles to perform numerical and symbolic computations.
  2. use technology appropriately to investigate and solve mathematical and statistical problems.
  3. write clear and precise proofs.
  4. communicate effectively in both written and oral form.
  5. demonstrate the ability to read and learn mathematics and/or statistics independently.
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. 
Disability Services:

The University of Tennessee provides reasonable accommodations (academic adjustments and auxiliary aids) to ensure equal access to educational content and university programs for students with disabilities.  Any student eligible for and requesting accommodations due to a disability must provide instructors with a letter of accommodation from Disability Services.  For additional information, please contact the Disability Services office at 209 Clement Hall, (731) 881-7605.