Mathematical Analysis I

Mathematics

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.

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

Example: State and prove the Bolzano-Weierstass Theorem.

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

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

- Critique mathematical writing of beginning analysis.

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

- 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.

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

- 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?

- 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.

MA 425-002 home page

Text and schedule | Course policies

Homework | Hints and answers

Test study guides | WebCT

Instructor's home page

Last modified Mon Jan 23 2006

Send questions or comments to schecter@math.ncsu.edu