#### Prerequisites & Bulletin Description

#### Course Objectives

The principal aim of Real Analysis II is for students to learn how to carry out a rigorous analysis of the convergence of infinite series of numbers and functions. They are expected to understand and apply the major theorems involving metric spaces.

Students learn to solve problems using the concepts of analysis. They present their solutions as rigorous proofs written in correct mathematical English. Students will be able to devise, organize and present brief (half-page) solutions based on definitions and theorems of analysis.

Students who successfully complete this course should be able to:

- Prove the Weierstrass M-test.
- Prove various tests of convergence of series.
- Prove Lindelof's Theorem.
- Prove the Heine-Borel Theorem.
- Prove the Extreme Value Theorem.
- Demonstrate their facility by solving problems that make use of the above theorems.

#### Course Requirements

Students will submit written homework on a regular basis. Homework assignments will require student to read the text or other sources, provide proofs or counter-examples of propositions, explore examples, and make conjectures. Beyond the work required in MATH 470, students in MATH 770 will be required to write a paper on an approved advanced topic that makes use of multi-variable analysis. MATH 470 and 770 are paired. Students who have completed one of the two for credit may not take the other course for credit.

#### Evaluation of Students

Students will be evaluated on their ability to devise, organize and present complete solutions to problems written in correct mathematical English. While instructors may design their own methods of evaluating student performance, these methods must include in-class examinations, frequent homework assignments and a final exam.

#### Course Outline

Topics |
Number of Weeks |

Infinite sequences and series of functions, uniform convergence | 4 |

Introduction to metric spaces, Heine-Borel Theorem | 5 |

Differentiable maps between Euclidean spaces; Derivative matix, Chain Rule, Mean-Value Theorem | 4 |

Inverse Function Theorem | 1 |

#### Textbooks & Software

*An Introduction to Analysis, *Fourth Edition, by William Wade.
*Elementary Classical Analysis, *Second Edition, by J. Marsden and M. Hoffman.

Submitted by: David Ellis and Yitwah Cheung

Date: September 19, 2008