Penn Math Math 202: Proving Things: Analysis Fall 2018

Faculty: Jerry L. Kazdan
    Telephone: (215) 898-5109
    email: kazdan AT
    Office Hours: Wed. 10:30-11:30   (and also by appointment) in DRL 4E15
TA: Man Cheung Tsui
    Telephone: (215) 746-7326
    Office Hours: Mon. and Wed., 5:30-6:30 (and also by appointment) in DRL 4C15.
    Web page:

This course focuses on the creative side of mathematics, with an emphasis on discovery, reasoning, proofs and effective communication.

We will study real and complex numbers, sequences, series, continuity, differentiability and integrability for functions of one variable, proving our way as we go, and enjoying a number of challenging problems.

This course is about thinking, attempting to understand. It it for students who enjoy thinking hard, even when completely stumped. Then understanding (if and when it comes) is all the more satisfying.

Intuition and computational skill will be essential in the discovery and presentation of your ideas.

The concept of a proof will be vital. There is nothing exotic about a proof. It is simply convincing someone else about your reasoning. You give different proofs to different people, depending on their background — and how well you know them. The first and most important person to convince is yourself.

Small class sizes permit an informal, discussion-type atmosphere, and often the entire class works together on a given problem. Homework is intended to be thought-provoking, rather than skill-sharpening. The evening labs will provide opportunity for all students to present their solutions at the blackboard, and to become comfortable and proficient at doing this.

To be successful in this course, you should be present for all class meetings and plan to take good notes.

Text: John P. D'Angelo and Douglas B. West Mathematical Thinking: Problem-Solving and Proofs, Second edition, Prentice Hall (2000). See also: Typos and corrections. [Note: There is also a newer paperback "Classical Edition", It is the *same* book.]

Some References

Course Grading

Prerequisites & Review Material: Some experience with calculus.
To remove rust from your background I suggest doing the problems from recent Math 103 Final Exams. Note that both Math 103 and 104 focus on techniques for solving standard calculus problems. This course will focus on basic ideas and proofs.

Problems to think about during August.
On the first day of class some students might be invited to present their solutions at the blackboard.

Symmetries of a Square
Big numbers
Axioms for a Field
Real solution of x2 = 2
Logic Notation
Newton's method for square roots.
Some Puzzles
TeX/LaTeX Stuff Creating documents having formulas.
ODE: Existence
MVT: How do you prove that?
The Fundamental Theorem of Algebra

Homework Assignments:

Exams: There will be three in-class exams, from 10:30-12:00 on

Tues., Oct. 2,   Thurs. Nov. 1,   Thurs. Dec.6   [No Final Exam]

You may always use one 3"×5" card with handwritten notes on both sides.
Exam 1: Original, condensed, solutions.
Exam 2: Original, condensed, solutions.
Exam 3: Original condensed, solutions.

Old Exams: 0