# Math 413 Analysis I

## Fall 2003

Instructor: David Marker
Call Number: 67387
Class Meets: 1200 MWF 220 SH
Office: 411 SEO
Office Hours: M 1:30-3, W 9-11, Th 10:30-12
Finals Week Office Hours T 12-3, Th 10:30-12, 1-3 phone: (312) 996-3069
e-mail: marker@math.uic.edu
course webpage: http://www.math.uic.edu/~marker/math413

#### Prerequisites

Grade of C or better in Math 215 Introduction to Advanced Mathematics, or consent of instructor.

#### Description

The main purpose of Math 413--414 is to rigorously revisit many of the ideas and results from Calculus I and II, including:
• Properties of the Real Numbers
• Sequences
• Elementary topology of the real line
• Limits and Continuity
• Differentiation
• Integration
• Series and Power Series
As time permits, additional topics may include metric spaces, differential equations and Fourier series.

Here is a detailed week-to-week syllabus.

#### Problem Sets

As in all advanced mathematics courses, homework problem sets are an essential part of the course. There will be weekly homework assignement. You may discuss homework problems with other students, but you must write up your solution independently. All proofs should be written neatly with complete gramatical sentences. Each problem should be submitted on a separate page. Late homework will be accepted only in exceptional circumstances.

• There will be two midterm exams. The best of your two scores will count for 30% of your final grade. The midterm exams will be on Friday October 17 and Wed November 26.
• The final exam will count for 40% of your final grade. Part of the final may be a take home exam. The final exam will be Friday December 12 at 8:00 am.
• Homework will count for 30% of your final grade. The three lowest homework scores will be dropped.

#### Bonus Problems

• Bonus Problem 1: The continued fraction expansion of the square root of 2. (Turn in by 10/3) Note: There was a typo on this problem.
• Bonus Problem 2: Every sequence has a stricly monotonic subsequence. (Turn in by 10/3)
• Bonus Problem 3 A sequence with subsequences converging to every possible limit. (Turn in by 10/3)
• Bonus Problems 4 & 5 Fat Cantor Sets, Limits of limits (Turn in by 10/31)typo corrected
• Bonus Problem 6 Exercise 4.3.9 (Turn in by 10/31)
• Bonus Problem 7 Exercises 4.6.4 and 4.6.5 (Turn in by 11/14)
• Bonus Problem 8 (Turn in by 12/5)