MATH 4900 Real Analysis Syllabus
Fall 2003

Instructor: Dr. Ron Smith
Office: 120 Science
Phone: 784-5283 (6473)

Office hours: MWF 10-11; TTh 10-12 & 1-3.

For a 20-minute appointment, just sign up on the sheet outside my office door or call me. You do not need to sign up in advance, but if you will please sign when you come, the time will be reserved for you. Also, if someone is in the office and you are signed up, please make your presence known!

Materials:

The required text is Understanding Analysis, by Stephen Abbott.

Scope:

Real Analysis is a course designed for students who wish to go on to graduate school in mathematics, or for students with a good calculus background who wish to gain a better understanding of the mysteries of the infinite. This is an introductory course, and our emphasis is on a solid introduction rather than a thorough grounding in the topic. Our aim is to firm up the student's understanding of the logical structure of limits with the end in view of exposing the student "to the tantalizing complexities of the real line, to the subtleties of different flavors of convergence, and to the intellectual delights hidden in the paradoxes of the infinite."

Expecations for the student:

Doing real analysis requires the student to train their intuition about sets, real numbers, functions, and convergence. We will pay more attention to the definitions and hypotheses that lie at the heart of our understanding of real numbers. Second, the student must learn to communicate in the language of mathematics. These are important skills that require practice. You are expected to attend all assigned class sessions, to read the book, to work on problems every day outside of class, and to ask questions both in and out of class.

Expectations of the Instructor:

My job is to coach you so that you can learn to read, do, and communicate mathematics. I will do my best to help you make sense of the mathematics you are involved in, to give feedback on your work, and I will try to make sure that you are developing your mathematical maturity.

Grades:

Grades will be based on a notebook of problems, and exams. All points will be added to determine your grade. You will need 87.5% for an A, 75% for a B, 62.5% for a C, and 50% for a D.
  1. Notebook (200). Each student will keep a notebook of problems. Each problem will be graded according to the 20 point rubric shown below. The average of all problems will be multiplied by 10 for the notebook grade.
  2. Exams (200) There will be a midterm and final exam, each for 100 points. The final is comprehensive, and will be held in 101 Science on Thursday, May 12, at 8:00 AM.

Incompletes

Incompletes for the course require signing a contract for making up work, and must be initiated by the student.

Academic Integrity:

Honesty is a prerequisite for being a competent person. If you copy solutions to problems from any source, you are required to acknowledge the source. This includes copying from friends or old homework/test files. Working together for inspiration and asking for hints is allowed on everything but exams. However, writeups must be your own. For more on this subject, see the college policy printed in the handbook.

Disabilities:

Any student who, because of a disabling condition, may require special arrangements in order to meet course requirements should contact the instructor as soon as possible to make necessary accommodations.

Rubric for grading problems/proofs

5 Outstanding4 Good3 Marginal1 Unsatisfactory
LogicLogic is valid. Assumptions explicit. Contradictions clearly stated and shown. Compututation is appropriate, complete. Conclusion drawn. Result is general.Logic is valid. Assumptions not explicit, or contradiction unclear. Computation is appropriate but not complete. Conclusion vague. Main argument valid; logic may be flawed. Computation is appropriate, but not necessarily correct. Question is partially answered. Main argument invalid. Computation may be inappropriate, or question is not addressed.
OrganizationQuestion is explicit. Ideas are linearly connected. No hand waving required. All variables named. Conclusion is clear and explicit. Question is explicit. Ideas are linearly connected. Little hand waving required. Variables named. Conclusion is clear but may be implicit. Question is clear. Ideas connected, but hand waving required. Variables may not be names. Conclusion missing or unclear. Question not stated. Ideas not explicitly connected--loosely or randomly strung together, or there is no identifiable structure. Variables not named. Conclusion missing or unclear.
Problem Selection
(If applicable)		Challenging; requires significant intellectual resources; potential for a new or different understanding. Requires above average intellectual work; complex computation, or modest computation with interpretation. Requires modest intellectual work; modest computation with minimal interpretation. Requires little or no intellectual work: e.g. Fill in the blank computation.
Documentation (If applicable)Sources, whether quoted, paraphrased, or summarized, are correctly cited. Minor errors in the in-text citations or bibliography; a reader can easily tell where sources are used. Occasional flawed or missing citation obscures the source being used. Sources are not cited or documented, and often stray from the proper form.

A Checklist for Writing Proofs.

  1. Clearly state the theorem you are trying to prove.

    Do not assume the reader knows what theorem you are proving. In complicated situations, you may start by proving a lemma or a special case of the theorem before coming to the main proof.

  2. Distinguish between the statement of the theorem and the proof.

    A good way to do this is to start the proof with the heading, "Proof:"

    GOOD:
    Prove: There are exactly three lines in the three point geometry.
    Proof: There are exactly three points ...

    BAD:
    There are exactly three lines in the three point geometry. There are exactly three points...

  3. Avoid "it".

    Give names to all the mathematical objects with which you are working, so that you never have to refer to one of them with pronouns like "it" or "them".

    GOOD: Given a line m and a point A not on m, there is exactly one line through A that does not meet m.

    BAD: Given a line and a point not on the line, there is exacly one line through the point that does not meet it.

  4. Draw a general picture.

    Whenever possible, draw a picture showing a model of the objects with which you are dealing in the proof. Usually, your picture will include the "set up" with all given objects labeled with the names used in the proof. Be careful to make your picture as general as possible so that you do not mislead your reader (or yourself) into jumping to unwarranted conclusions.

    Example: A picture of arbitrary lines in a finite geometry should not include a point of intersection unless two lines always intersect, and then only after you have proved that fact.

  5. Clearly denote the contradictions.

    In a proof by contradiction, state what has been assumed, and clearly denote the contradiction.

    GOOD:
    Axiom 2: Two lines are on at least on point.
    Axiom 3: Two points are on at most one line.

    Two arbitrary lines m and n are on at least one point, A, by Axiom 2. Suppose m and n are also on point B. Then A and B are on two lines m and n, contradicting Axiom 3.

    BAD: Two lines m and n are on at least one point by Axiom 2. If m and n lie on another point then there is a contradiction.

  6. Conclude.

    Don't leave the reader hanging in suspense. State clearly what has been shown.

  7. Write to communicate.

    Use complete sentences with correct grammar, spelling, and punctuation.

  8. Write linearly.

    Avoid arrows directing the flow of the proof. When you must break the linear flow of thought, label the important lines and refer to the labels.

    E.g. Subtracting equation (1) from (3) gives ...

  9. Write logically.

    Avoid logical fallacies, such as using the converse or the inverse instead of the contrapositive. Avoid circular arguments.

  10. Be complete.

    Cover all the possibilities.

Last Update: January 24, 2005
Ronald K. Smith
Graceland University
Lamoni, IA 50140
rsmith@graceland.edu