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:  E-mail is the standard method for us to communicate if the school is closed for an extended period of time.  I will use your e-mail as stored in Banner, usually an account.
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 will try to do a great deal of board work (student proofs in class on the board, often 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 241/314, then you will remember my slogans: "learn to learn," and "write to be read."  They will be emphasized here as well.
Discrete Mathematics (Math 314 = 241) 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.
The course grade will be a weighted average of the homework (17%), tests (53%), final (17%), board-work and project (13%).  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.
(To be determined.)
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. 
Principle of Real Analysis ["Baby Rudin"] 3ed McGraw Hill.  ISBN: 978-0070542358 (0-07-054235-X) hardback or 978-0070856134 (0070856133) paperback.