# MATHEMATICS 501-Fall, 2009

# Real Analysis

**Carver
2 (basement)**

### Instructor: Howard A. Levine , Department of Mathematics, 410
Carver

### Office Hours: MWF 9:30-10:50 or by appointment

### Textbook:
Principles of
Real Analysis by Walter Rudin

Suggested alternate reading (in no particular order)

Manfred Stoll, Real Analysis

Russel l A. Gordon, Real Analysis, A First Course

William R. Wade, An introduction to Analysis

Robert Bartle and Donald Sherbert, Introduction to Real Analysis

### Grading: ** Homework: 40%, Midterm 20% and Final: 40%
**

The grade will be
determined from, homework, (200 pts.) the mid term, (100 pts.) and the
in class final (200 pts.) . (There will be seven such
assignments
each consisting of roughly ten problems.) Problems will be of
varying difficulty and will be weighted accordingly and scaled to 200
pts at the end of term (40% of the grade). You may consult with
your fellow students but the solutions you turn in must be
written in your own words!

### NOTE:
The homework problems will generally be a subset of those
problems found at the end of each chapter in the text. A
subset of the problems from each set will be graded.

While I do not require the homework to
be typed up in tex or some other text editing program, I do expect them to
legible and written in clear English,
otherwise I will penalize you
25% .
**Course Lectures: I will follow the
textbook, but not slavishly.
You are expected to read and understand the material
and
fill in the gaps that I will sometimes (deliberately) leave in the
proofs I give in class or that the text omits.**

### Syllabus:

1. Chapter 1, Real and
Complex Number Systems, all sections and the appendix.

2. Chapter 2,
Basic Topology, all sections

3. Chapter 3,
Numerical Sequences and Series, all sections.

4. Chapter 4,
Continuity, all sections.

5. Chapter 5,
Differentiation, all sections.

6. Chapter 6,
The Riemann-Stieljes Integral all sections.

7. Chapter 7,
Sequences and Series of Functions, all sections.

Homework
problem sets are given below. . Due dates for each set
are at the top of each page. Late homeworks carry a 20%
penalty.

### Problem sets will be added to the list from time to time.
Check back frequently.

Problem Set I, Page 22 of text : 2, 3, 5, 6, 7, 8 ,
13, 16, 19

Also show that the set of rational numbers whose
cube is
smaller than 3, defines a Dedikind cut.

Set I is due on September 7, 2009.

Problem Set II. Page 43 of text: 1, 2, 6,7, 8, 11, 12, 17, 21, 23,. Set II is due on September 25, 2009.

Problem
Set III. Page 78 of text. 3, 4, 6, 7, 8, 9, 10,
11, 12. Set III is due on October
14, 2009

Problem Set IV Page 80 of
text. 13, 14 (a-d only) 16, 17, 18, 23 Set IV
is due on October 23, 2009

Problem Set V Page 98 of
text. 3, 4, 6, 7, 8, 14, 18, 20, 21, 23. Set V
is due on November 4, 2009

Problem
Set VI Page 114 of text . 2, 6, 8, 9, 11, 13,
14, 15, 25, Set VI is due on November 16, 2009

Problem
Set VII Page 138 of text. 3, 4, 5, ,6,
8, 9, 10, 13, 15 Set VII is
due on November 30, 2009

Problem
Set VIII Page 165 of text . 2, 3, 4, 6,
7, 9, 11, 15, 16, 18 Set VIII is due on
December 11, 2009

# MIDTERM EXAMINATION
INFORMATION:

Midterm examination will be on October 19. It will cover the material from Chapter 1 through Chapter 3 (except as noted in a recent e-mail).

# FINAL EXAMINATION
INFORMATION:

The Final exam will be based on Chapters 4-7.
Carver 2 | Wednesday | Dec. 16 | 9:45 a.m. - 11:45pm |

** **