Preface To the Student
Analysis is an area of mathematics, just as Algebra, Geometry, and Topology are areas of mathematics, and is usually defined heuristically or not at all. In your calculus sequence you learned about the topics of limits, continuity, differentiability, and integration at an introductory level. In this course, we will follow the same order that you followed in calculus, but we will spend more time on the mathematical structures than on the application of the concepts. We will define each concept rigorously and present material that you will recognize from calculus such as the Extreme Value Theorem, Mean Value Theorem, Rolle's Theorem, and the Fundamental Theorem of Calculus. From here, we will explore the notions of uniform continuity, uniform convergence, the existence and uniqueness of solutions to differential equations, and a bit of measure theory.
These notes are intended to be virtually self-contained, only assuming basic understanding of the real numbers. If you work through the first fifty problems independently, then you have mastered the topic that most universities would cover in a course titled “Real Analysis” or “Advanced Calculus.” Universities typically offer such a course after a “transitional” course intended to move students away from working problems and toward a theoretical viewpoint of the subject.
The goal is to solve all the statements labeled as “Problems,” “Theorems,” or “Lemmas.” The titles don't necessarily represent the level of difficulty because the real work in proving a theorem may have been done in a lemma or problem. A handful of the problems or theorems are labeled with (CA) which implies they require the use of the Completeness Axiom.
The remainder of this introduction is important only to those taking the course from me for credit.
All work presented or submitted is to be your own. You are not to discuss any problem with any one other than me, nor are you to look to any other reference such as a book or the internet for further guidance.
Grading for the course will be no less than the average of three grades: your presentation grade, your submission grade, and the average of your midterm and final exam grades. Anyone who is regularly presenting material at the board will certainly have adequate work for good submission grades and thus will likely do well on the midterm and final. The midterm and final grades are opportunities for those who do not regularly make it to the board. However, it is my experience that those who do not work toward successful presentations rarely do well on the midterm and final. Thus, I emphasize that the goal of the course for each student should be well prepared, well presented problems at the board. You will know your grades on submissions and tests. My policy for your presentation grade is:
D = the student made it to class every day, was attentive and alert, and his or her cell phone never rang
C = requirements for D plus made a few successful presentations
B = requirements for C plus made numerous successful presentations
A = requirements for B plus presented some truly impressive problems
Turn-ins. You must turn in exactly one “new” problem each week. A “new” problem means one that you have not turned in before. This problem should be neatly written and double spaced. You should label this problem with TURN IN at the top of the page along with your name, the problem number and the problem statement.
Grading for TURN IN assignments will be based on the following scale.
A = This is a correct proof.
B = You know how to prove the theorem but some of what you have written is not correct.
C = You have a mistake in your work or I do not understand what you have written, but I believe you have a good idea.
D = There is at least one major flaw in your argument.
Please understand that the purpose of the TURN IN assignments is to teach you to prove theorems. It is not expected that you started the class with this skill; hence, some low grades are to be expected. Do not be upset - just come see me.
Resubmissions. If you receive a grade of less than B, you may resubmit a TURN IN problem on the following week and I will average the two grades. You still must turn in a new problem as well and you only get one chance to resubmit. Please write RESUBMIT at the top. Feel free to come see me anytime if you do not understand my comments. It is expected that a certain amount of time in my office will be required to help students. I prefer to give guidance in my office rather than in class because this allows me to tailor the hints to the person who needs help, but I will always answer questions in class as well.
Boardwork. If you have solved a problem that is about to be presented at the board or you feel you have made significant progress on a problem that is about to be presented then you may opt to leave the room for the presentation. In this case, you may turn in a write up of this problem for credit as BOARD WORK. You must write BOARD WORK at the top of the page. There is no limit on the number of BOARD WORK problems you can submit.
Last Comments. Be sure that everything you turn in has your name, problem number, and problem statement on it. Be sure to double space and write either TURN IN, RESUBMIT, or BOARD WORK at the top.
I reserve the right to vary from these policies and probably will.