MthSc 453, Summer I 2012
# MthSc 453, Summer I 2012

"You see this little hole? This moth's just about to emerge. It's in
there right now, struggling. It's digging it's way through the thick
hide of the cocoon. Now, I could help it - take my knife, gently widen
the opening, and the moth would be free - but it would be too weak to
survive. Struggle is nature's way of strengthening it."

--Locke (*Lost*, 2004)

### Resources

• Class
Syllabus

• **Text**: *Princples of Mathematical Analysis*, by
Walter Rudin. This book, affectionately called "Baby Rudin" was
published in 1976 and to this day remains the gold standard of
introductory real analysis texts. For an idea of why it is the best,
read
this friendly review.

• Francis
Su of Harvey Mudd
College taught Real Analysis in 2010 (using Rudin's book) and the
lectures were videotaped and
put on
YouTube. Professor Su is one of the finest teachers in the
country, and was the first recipient of the
Henry
L. Alder Award for Distinguished Teaching, a national award given
out by the Mathematical
Association of America. These lectures are *not* a substitute
for attending class, but are a great supplement. Also check out
Rudinium.

• Good mathematical writing is *difficult*. It takes hard
work and practice to acquire this skill, and developing it is
of the primary goals of this class. Francis Su has written a very
nice 4-page essay on the
topic.

• The late Evelyn Silvia taught real analysis out of Rudin at
UC Davis. She wrote a set
of companion notes to go along with Rudin's book.
### Homework

Homework 1: pdf |
tex. Due
Friday, May 18th at 4pm.

Homework 2: pdf |
tex. Due
Tuesday, May 22nd at 4pm.

Homework 3: pdf |
tex. Due
Friday, May 25th at 4pm.

Homework 4: pdf |
tex. Due
Tuesday, May 29th at 4pm.

Homework 5: pdf |
tex. Due
Friday, June 1st at 4pm.

Homework 6: pdf |
tex. Due
Tuesday, June 5th at 4pm.

Homework 7: pdf |
tex. Due
Friday, June 8th at 4pm.

Homework 8: pdf |
tex. Due
Tuesday, June 12th at 4pm.

Homework 9: pdf |
tex. Due
Friday, June 15th at 4pm.

Homework 10: pdf |
tex. Due
Tuesday, June 19th at 4pm.

• Midterm 1
Solutions

• Midterm 2
Solutions

### Lecture notes

Lecture 1:
Construction the rationals

Lecture 2:
Properties of Q

Lecture
3-4: Constructing the reals

Lecture 5:
Complex numbers

Lecture 6:
Principle of induction

Lecture
7-8: Countable and uncountable sets

Lecture
8-9: Metric spaces

Lecture 10:
Open and closed sets

Lecture
11-12: Compactness

Lecture 13:
The Heine-Borel theorem

Lecture 14:
The Cantor set and connectness

Lecture 15:
Convergent sequences

Lecture 16:
Subsequences and Cauchy sequences

Lecture 17:
Completeness

Lecture 18:
Series

Lecture 19:
Convergence of series

Lecture
20-21: Limits and continuity

Lecture 22:
Uniform continuity

Lecture 23:
Discontinuous functions

Lecture
24-25: Differentiation