Mathematics

### Goals of the Course

In MA 425, the student studies sequences of real numbers; topology of the real line; continuity, differention and integration of functions of one variable; uniform convergence; and infinite series. Unlike a calculus course, MA 425 focuses on rigorous definitions and proofs.

At the end of the course, the student should be able to do the following things.

1. Reproduce clear and precise definitions, theorems, and proofs of beginning analysis.

Example: State and prove the Bolzano-Weierstass Theorem.

2. Identify properties of objects of the types studied in beginning analysis.

Example: Which of the following sequences of functions converge uniformly?

3. Critique mathematical writing of beginning analysis.

Example: Read a "proof" that a specific sequence converges and find mistakes, unclear parts, and extraneous parts.

4. Apply the definitions and theorems of beginning analysis to prove facts about examples and to prove new theorems.

Example: Prove that a specific sequence converges.

Example: Suppose the sequence a_n approaches a and the sequence b_n has b as an accumulation point. Prove that the sequence a_n b_n has ab as an accumulation point.

5. Write proofs that are (a) correct, (b) clear, (c) well-organized, and (d) without extraneous material.

6. Explore examples and make conjectures about properties of examples and general facts.

Example: Suppose the sequence x_n is defined inductively by x_{n+1} = x_n/(1+x_n). For which x_0 does the sequence converge?

7. Explain intuitively, in writing and speaking, the concepts of beginning analysis and their significance, at a level appropriate to other students in the class.

Example: Explain with the aid of pictures why a Cauchy sequence should have a limit.

Example: Explain what strange things can happen when a sequence of continuous functions converges pointwise but not uniformly.