Math5615F05.

Math 5615H: Honors: Introduction to Analysis I. Fall '05

Last changed: January 1, '06

The class meets at 12:20pm MWF, in VinH 20.
Instructor: Max Jodeit ("Yodite") VinH 258, 5-3855, jodeit@math.umn.edu
Text: Principles of Mathematical Analysis by Walter Rudin,
Third Edition, ISBN 0-07-054235-X

Office Hours: 10:30am - noon, MWF; otherwise, by appointment -- or call 5-3855 to see if I'm in and available.

News and Announcements will go here

Link to the papergrader's 5615H page
Links to PDF documents

NEWS Jan 1: Here's a link to Math 5616H in Spring '04

Math 56-6H, Spring '04 1/1/06

NEWS Dec 28: I apologize! Somehow, my Dec 23 NEWS did not get
posted!

NEWS Dec 23: The 5615H grades for Fall '05 have been submitted to the
Registrar. They should be available by 11:15pm Dec. 24. I hope you all have a happy
time during the break! You may look at your Final Exams after Jan. 17 (and I recommend
that you do).

NEWS Dec 20: The Final Exam is tomorrow, Wednesday Dec 21, from 4 to 6pm,
in the regular classroom, VinH 20.

Questions on the Final:
Q1: Can we assume that questions on
Tests 1 and 2 will not appear on the Final Exam?
Ans: No. In fact, working Test Problems is a good way to prepare.

NEWS Dec 13: A solution of Special Problem 9 is now in place.
Tomorrow will be the last class before the Final, and will also be Course
Evaluation Day. Please bring a #2 pencil along!

NEWS Dec 6: Assignments have been updated: Assignment 12;
they might not all be scored since they need to get back to you Wednesday.
Reminder: the Final is on Wednesday Dec 21, 4pm to 6pm, in VinH 20.

NEWS Nov 27: Assignments have been updated: Assignment 11;
please read the Nov 26 NEWS.

NEWS Nov 26: Please bring your Special Problem 6 paper
to show me; I lost the list of scores! Also, your "grade so far" has been (first draft)
done, so if you bring you SP 6 to my office, I can let you know that number...

NEWS Nov 22: Assignments have been updated: TWO Special Problems,
numbers 8 and 9, due the same day!

NEWS Nov 18: Assignments have been updated: Assignment 10,
Due November 28, NOT Nov. 21! Special Problem 8 will appear over the weekend.

NEWS Nov 12a: Assignments have been updated: Assignment 9
(Chap 3, # 1, 2, 4), Special Problems 6 and 7.
A note has been added, that gives a correct example of an incomplete
metric on the real numbers that nevertheless determines the same open sets
on \real as does the usual metric.

NEWS Nov 8a: Another question has arrived. It's "Q4," below.
Note that the incompleteness example is not covered on Test 2.

NEWS Nov 7b: I used the wrong metric in class today for the
incompleteness! The metric I used in class makes \real into a complete
metric space. This was shown in an email from jacek@socsci.umn.edu
wiseman@econ.umn.edu cslavik@econ.umn.edu barnette@econ.umn.edu
troshkin@econ.umn.edu. I've asked their permission to post their file.

Permission to post their file has been received.

I should have used a different metric, the one based
on the map between [-1, 1] and \bar R! That is,
d(x, y) = | x(1+|x|)^{-1} - y(1+|y|)^{-1} |. With this metric,
x_n := \sqrt{n} is a non-convergent Cauchy sequence.

NEWS Nov 6,7,8:
Questions on Test 2:
Q4: To show that a sequence does not exist (or diverges), can
we use (in Test 2) the theorem that a real valued sequence converges
iff its limsup=liminf?
Ans: No. That has not been covered in class yet.
Q3: Should we use Rudin's definition of a subsequence, or should we
use the one from class (the one from class involved defining a subsequence
as a composition of functions)?
Ans: Use either one. This applies to all definitions and statements of
Theorems. In particular, Rudin's version is always acceptable, even if it was not
given in class.
Q2: What is the (sufficiently for the test) complete proof of
1) set {1/n : n is natural} has a limit point 0?
2) sequence 1/n converges to 0?
Ans: I will only give hints for questions like this:
to give a "complete" answer would defeat the purpose of asking
you to think, and would serve memorizers to the disadvantage of
everyone else.
Hints: 1) (1/0 is undefined!) show that every (-r, r) contains a
fraction 1/n.
2) Show that every (-r, r) contains all but finitely many of the fractions 1/n.
Q1: Are we responsible for the proof that the rational numbers are dense
in the set of real numbers (usual metric), on Test 2?
Ans: No, not on Test 2.

NEWS Nov 3: A note on Theorem 3.7 is now on the Web.

NEWS Nov 2: Test 2 will be Wednesday Nov 9. As for the last
test you may ask questions about Test 2 by email. The question and its
answer will appear here. Coverage will begin approximately at 2.24 and
continue into Chapter 3, thru 3.20, maybe.

Old News. 11/12

The links below are to PDF documents. They require Adobe's Reader.

Syllabus and Assignments Assignments 12/6 Syllabus 9/6

Special Problem 9: a solution 12/13

An incomplete metric on \real 11/12a

A note on Theorem 3.7 11/3

Special Problem 3: a solution 10/16

On Inductive Sets 9/15

Axioms for the Real Numbers 9/6

On Complex Numbers

Sets, variables and quantifiers

Logic and sets