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Math 104 Fall 2006
## Welcome to math 104 for spring 2011.

Math 104 is a course of central importance. In it you will
learn the careful mathematics which justifies the intuitive
arguments of calculus. You will come to grips with the epsilons
and deltas you flirted with in calculus and find out the huge
difference between the notions of continuity and differentiability.
You will learn exactly what is meant by a real number and go
into depth about the basic properties of the real number system.
You will also learn how to write careful proofs and strive for
elegance in your writing of them.
All this does not come easily and it is a course you will find
demanding, probably more demanding than any other course you ever
take.
The text for the course will be Ross "Elementary Analysis, The theory of calculus."
I have not used this book before in teaching 104 so it will be an adventure
for all of us.
The class is too large for email to be practicable. Please communicate
with me at office hours or briefly before and after lectures.
Homework is probably more important for this course than for any other
course and I will assign homework weekly. A lot of the homework problems
will require a considerable amount of reflection so it will be hopeless
to start on them the night before they are due.
The grade in the course will be based on the homework, one midterm
and the final.
I will hold regular office hours, tentatively Tu 2-3 W 3-4 and by appointment.
The appointment should be made right before or after the lectures. These office
hours must be used appropriately. They are not for me to do the homework for
you or to repeat the lecture for you. Let us say that you may ask me a question
but only if you have already thought about it for at least half an hour
beforehand. Often you will find that simply by making yourself phrase the question correctly you will be able to answer it wihout help.
## HOMEWORK

Homework (due Thursday 27 January)
**page 25**Do 4.1,4.2,4.3 and 4.4 for (b)(j)(l)(n) and (t) of 4.1.
**page 26**:4.8,4.12,4.14,4.15 and 4.16.

Homework (due Tuesday 1 February)
**1) Show that in a complete ordered field every element is the sup of
the set of rationals less than it. (this you have actually already done
in the first homework, no need to do it again if you got it last time)
****2) Show that in a complete ordered field a non-zero element is positive
iff it is a square.
****3) If F and G are complete ordered fields and f and g are nonzero maps
from F to G satisfying f(x+y)=f(x)+f(y) (same for g) and f(xy)=f(x)f(y)
(same for g), then f and g are equal bijections.
**** 8.1(b),8.3,8.7(a)(c),8.10
**

**
**## Last office hours

I will hold a review in the usual lecture room(I hope) on
Tuesday 3 May 12:30-2. Then office hours the same day 2-3 and 4-5.
Then I'll be in my office on the day of the exam, Thursday 12 May,
from 10am to 11:45 am.

**
**Homework
due Tuesday 8 February

Proof of the result in class about the equivalence of completeness of an ordered field and a property of monotone sequences.

**
** Proof that a Cauchy sequence of reals converges.

Equivalence relations and partitions.

Construction of the reals from the rationals.

Schroeder-Bernstein, Prof Woodin's proof.

A student's unofficial lecture notes.

Homework
due Thursday 17 February

Homework
due Thursday 24 February

Cardinality.

Definition of the Cantor set.

Homework
due Thursday 3 March

Homework
due Thursday 10 March

Homework
due Thursday 17 March

Homework
BY POPULAR DEMAND due only Thursday 7 april

Homework
consisting of sample T/F questions for midterm.
Not to be handed in.

Homework
due Thursday 21 april (typo corrected in q.7)

Homework
due Thursday 28 april

Last Homework
not due.

A students homework solutions.
(Unofficial.)

The set of
scores in the midterm.

The set of
scores in the makeup midterm.

Midterm solutions.

Makeup Midterm solutions.

Metric space concepts.

Compact iff sequentially compact.

Professor Morrison's notes.

My 104 final from last time.

Midterm instructions.