Course Objectives
Real Analysis concerns functions whose domain and range are subsets of real numbers.  The important concepts of calculus, namely  limits, continuity, derivatives, and integrals, play a central role.  However, in contrast to your past study of calculus, where the primary focus was on understanding and applying procedures, in this course the main focus is on the theory.  Your goal will be to understand deeply what the real number system is, how important concepts are defined, what truths are known about them, and how we know they are true.  Making the comparison in another way, in calculus you were expected to work out answers to quantitative questions by calculation and manipulation; here you will be expected to work out answers to theoretical questions by experimental investigation and logical deduction.

There are therefore two main objectives of this course:
  1. learning the theoretical foundations of real analysis, consisting of definitions, theorems, and examples, and
  2. successfully applying the techniques of analysis to answer (with logical proofs) theoretical questions about functions, limits, and so on.
These encompass
At the end of the course you should know the main definitions and the main theorems for the topics we cover,  why the definitions and theorems are formulated in the way that they are, and what they mean. You should also know the proofs for the main theorems (or at least outlines of the key ideas for some proofs), as well as the kinds of reasoning that go into these proofs. 

Prerequisites
Math 503 is a prerequisite for this course.  Students will be assumed to know the names, notations, and properties of important number systems (natural numbers, integers, rational numbers, and real numbers), basic properties and operations of sets and functions, the logical structure of definitions and proofs, including correct use and interpretation of quantifiers, and proof schemes such as proof by induction, direct proof, proof by contradiction, proof by contrapositive, and if and only proofs.    A separate handout provides a sample of statements you should be able to prove.

Grades
Grades will depend on exams and a portfolio of course work. The portfolio is required, and counts for 40% of your grade. You will be adding work to your portfolio throughout the semester, so it is important that you understand at the beginning what is expected. See a separate handout called Course Portfolio.

Tentative Schedule

A tentative schedule showing what topics are planned for each class meeting is available here.


Makeup Policy
If you are forced to miss an in-class exam for reasons beyond your control (such as an illness, family emergency, etc.), a makeup may be arranged, but ONLY if I am informed in advance. I will NOT approve requests to reschedule an exam for reasons of convenience. For example, if you plan to travel during a school break, that is not a valid reason to reschedule an exam. Similarly, avoidable conflicts for recreational, entertainment, social, or work activities are generally not valid reasons to miss an exam. You have received a schedule indicating the dates of the exams; please plan other activities around them.

Attendance Policy
Class participation in this course is important. Although I will not keep track of days you miss class, or impose specific penalties for missing class, I do expect you to attend and participate in each class meeting, unless you are ill or have unavoidable conflicts.

Homework
Homework will be assigned for every topic we cover.  You should make it a practice to work on homework after every class.  Assignments will be posted online and announced via blackboard and email.  .

One or more problems in each set of problems to hand in is designated as a plus problem (marked with a + symbol on the assignment sheet.  These problems have a special significance as explained in the Portfolio handout. For one thing, all plus problems will be collected and graded twice - once as part of a daily assignment, and a second time as part of a collection of plus problems. Consider the first time you hand a plus problem in to be a rough draft, and use the comments you receive to correct and polish your presentation.

In some cases you may not complete an entire problem set, either because you do not understand all of the questions, or because you run out of time. In all cases, hand in as much of the assignment as you have completed when it is due. Some of the students have had classes from me in the past, in which I was pretty lax on homework deadlines.  Because of my other obligations, this semester I am going to be pretty strict on the deadlines.  Late work will only be accepted in cases of illness or genuine emergency situations.  Since the problem sets will have a signficant impact on your grade, it is important to complete and hand in on time as much of each assignment as possible. Any problems you are unable to complete by the deadline should be completed as you have time and added to your portfolio.  But I will not collect or grade these problems.

Collaboration
You may work together on some problems, but you should work alone on a significant part of each assignment.  When you do work with others, observe the following guidelines.  First, each student should be actively involved in working on problems.  It is not of much value to copy the work of others, nor does it help someone to copy your work.  Second, each student should write up his or her own version of the solution to each problem.  It is fine to work together to understand what method to use, or how to approach a problem, but when it comes to actually writing up the solutions, work separately.  Your solution to each problem should be something you understand for yourself and can explain in your own words.  This is the most effective way to use the practice problems to help learn the material.

You should not get help from other students (or anyone else) when you develop final drafts of plus problems.  Please consult only with me for any questions that you have on these final drafts. 

The guidelines and rules for collaboration are explained in greater detail in a separate handout on Academic Integrity Rules.

FORMAT: Homework should be done on 3 ring loose leaf binder paper. Leave a wide margin on the left side of the page for any comments I want to make. Do not jam all of your work together, leave space between successive problems. Please staple the pages of your assignment.  Here is a sample showing the preferred format.  This format should be used for both regular assignments and final drafts of  plus  problems.  A special formatting requirement for final drafts is that each problem should begin on a separate sheet of paper, and should be clearly labeled. A clear statement of what is to be proven should appear first, followed by your solution. You may wish to organize your work by proving lemmas that you can cite in the main proof. In this case, each lemma should have a clear statement and proof separate from the main question.

Your work will be easier to read and easier to correct if you prepare it using a word processor.  This is particularly encouraged (but not required) for plus problems.  Suggestions about mathematical word processing are provided here.

For work prepared with a word processor, it is even more important to leave space for my comments.  Be sure to leave a generous margin on the left, and also leave some extra space after each problem.