Welcome to math 260 for fall 2014.
Math 260 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
The text for the course will be Gaughan "Introduction to analysis" Fifth edition.
The book is well written but does not go into as much depth as I would like.
I include below a link to the notes for this course last time I gave it,
at Berkeley. The notes were taken by a student in the course and are thus a bit
raw but should be valuable.
Notes from previous course.
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. In fact what I want you
to do is write up your solutions then wait at least 4 days without looking
at them. Then read over them again and see if you can understand them yourself
before handing them in.
The grade in the course will be based on the homework, two midterms
and the final which will be weighted approximately as final, 50%, each mid
term 20% and the homework 20%. The astute among you will notice that this
does not add up to 100%...
I was asked to provide a syllabus so here it is::
Math 260. Introduction to analysis.
Exisence and uniqueness of the real numbers. Sequences. Metric spaces.Open and closed sets. Continuity, Compactness and connectedness. Cantor sets.
Differentiability. Mean value theorem. Taylor's formula. Convergence of series and series of functions. Stone-Weierstrass theorem. Darboux sums and
I will hold regular office hours, tentatively Monday 2-3 Wednesday 3-4 Friday 2-3 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.
First MIDTERM will be on Friday September 26
first homework, due Monday 25 August
selected solutions to first homework, due Monday 25 August
second homework, due Monday 1 September
Main ideas of selected solutions to second homework
third homework, due Monday 8 September
First MIDTERM will be on Friday September 26
fourth homework, due Monday 15 September
CONSTRUCTION OF THE REALS FROM THE RATIONALS
fifth homework, due Monday 22 September
WHAT THE FIRST MIDTERM MIGHT LOOK LIKE
sixth homework, due Monday 29 September updated version!!!!
seventh homework, due Monday 6 October
Notes on compactness, Heine Borel and sequential compactness
eighth homework, due Monday 13 October
Schroeder Bernstein dynamical systems proof
homework 9 due Monday 20 October
Second MIDTERM will be on Friday November 7
Definition of the Cantor set
Uniform continuity, pointwise and uniform convergence, examples
In an uncharacteristic bout of magnanimity I am not assigning any
homework due Monday 27 October
homework 9 due Monday 3 November
assignments for writing up selected homework exercises
The second midterm, on Friday (7 November) will cover the course material
up to and including last Friday(31 October).
homework 11 due Monday 10 November
Students' solution of problem describing all open subsets of the reals
Students' solution of problem on rearrangements of sequences
homework 12 due Monday 17 November
homework 13 due Monday 1 December
A link from Win containing a dialogue about why the reals are necessary
Students' solution of problem on definitions of connectedness
Students' solution of problem on automorphisms of the reals.
homework 14 not due
metric space concepts
A nice discussion of the Cantor set found by Sunny (much more info than required in the course)
Final from a previous time I taught essentially the same course
Students' solution of problem on expressing reals in binary, page1.
Students' solution of problem on expressing reals in binary, page2.