Real Analysis I

Professor:	Dr. Douglas Anderson
	Ivers 234D
	299-4453
	e-mail: andersod@cord.edu

Office Hours: Tuesday and Thursday 1-4; other times by discovery.

Purpose: The purpose of this course is to revisit the essential issues of the calculus: sets, numbers, sequences, limits, continuity, differentiation, integration, and series, placing them on a rigorous foundation based on definition, theorem, and proof. To accomplish this goal, every exercise will involve the method of formal mathematical proof, including proof by mathematical induction, direct proof, epsilon-delta proofs, proof by contradicition, and if-and-only-if proofs. Ultimately we believe the clarity and precision of rigorous, analytic proofs train and discipline our thinking in a way that leads toward a more beautiful mind.

Text: Introduction to Analysis, 5th edition, by Edward D. Gaughan

Homework: Homework will be assigned each day for your development toward mastering the material through mathematical rigor, and will be due two class periods later. Each will be worth 30 points. I will drop your four (4) lowest homework scores before calculating your overall homework grade. Homework that is turned in should be neat and organized. Do not turn in messy papers with erasures, cross-outs, or restarts; work on scratch paper first, and then write up a nice, clean version for submission. This will take time. You will not only be graded on correct explanations but also on the neatness, organization, and clearly delineated steps used to derive your proofs. The use of correct English grammar and spelling amidst the mathematics is expected, within properly complete sentences.

Exams: There will be three (3) unit exams, plus the final exam.

Dear Cobbers: I have been asked to write you this letter as a bit of an introduction to Real Analysis, a class that I believe is the most difficult class in the mathematics curriculum. Over the course of this class, you will take the most basic facts of mathematics (the existence of numbers and a few theorems that you've known since third grade), and take those to lay a groundwork for much of the mathematics you've been using for the past several years, forcing you to question simple facts that you've never even thought about before. Real Analysis is a proof-based class -- very rarely will you be working with actual numbers. For this reason, I highly recommend that all math majors take Real Analysis, despite the fact that it is not officially required. Any of you that intend on going to graduate school in a math-related field will need a high mark in this class (along with Modern Algebra) to impress the committees looking at applications. Even if you're not thinking of going on to more school, Real Analysis will allow you to look deeper into the field of mathematics, give you more experience with writing proofs, and teach you a few unusual things that you won't expect to learn from this class. However, if you're just trying to coast through and get the required classes to graduate, this is probably not the class for you. To be a successful student in this class, you will need to learn a new method of studying mathematics. Rather than going over problems and working through them over and over, I've found the best approach is to memorize the theorems, look over the proofs in the book and those done in the homework to see some of the logic behind them, and use a little creativity on test day. While "using creativity on test day" sounds a little scary right now, I'm sure that as you work on your first few assignments, you'll understand what I mean -- many of these proofs require you to use multiple theorems, sometimes in unusual ways, to properly prove the statement at hand. Also, while I wasn't afforded this opportunity (as I took the class independently), forming study groups would be extremely helpful for this class, so do it early -- don't wait until after the first exam. Ultimately, while Real Analysis won't be easy, it will be a very rewarding class for you to take as you learn how the groundwork of mathematics actually works, gaining more experience writing proofs and reading dense mathematical writing along the way. I wish you all the best of luck, and be patient with it. It will be worth it to you at the end of the semester. Sincerely, Doug Anderson Class of '07.

Grading:		Points			Scale
	Homework	200			A 90%--100%
	Exams(3)	300			B 80%--89%
	Final Exam	200			C 70%--79%
					D 60%--69%
					F Below 60%