I will be in my office until about 3pm on Thursday Dec. 18 and will then be out of town, and without e-mail access until after Christmas.
Thanks for being such an agreeable group of people, and good luck in your future mathematical and non-mathematical endeavors.
Bruce Reznick
The TA for the course is Mr. Hua Tao. His office hours will be Tuesday 5:00 - 6:00 and Thursday 4:00 - 5:00 in 155 Altgeld Hall.
LAS and Engineering students enrolled in this class are permitted to drop this course without academic penalty and without petition through Fri. Nov. 21. See Sue Woodside in the LAS office, or the Engineering College Office, 206 Engineering Hall, as appropriate. After Nov. 21, any request to drop will be considered a late drop request.
A student writes: "I had a question regarding our homework. In some of my prior math classes, the rule has been that if a proof of a theorem was in our book, we could use the theorem in our homework withour rewriting the proof, however, in other courses I have been required to recopy the proof of the theorem into my proof. I was wondering what the policy is for this class." My reply: "You may quote any theorem or example from class or the book ... PROVIDED that it has been proved there. It is not acceptable to quote an unproved homework problem as a step in proving an assigned homework problem!"
A question came up in email about the false induction. Suppose P(n) is the statement that all sets of n objects have the same color. It is evident that P(1) is true. To show P(n) => P(n+1), I argued that if you line up the first n objects (which must be the same color by assuming P(n) is true) and then line up the last n objects, which must also be the same color, then because of the overlap, the n+1 objects must all be the same color. That is, assuming P(n) is true, we conclude that P(n+1) is true.
The fallacy is that in the implication P(1) => P(2), there is no overlap. I mentioned in passing that if P(2) were true, then the induction argument was valid, and we had that P(n) is true for n = 2,3,4,... The point here is that if P(2) is true , then any two objects are the same color, and this clearly implies that any set of n objects is the same color. This is not a huge point, but I'd hate to see any confusion about it.
F 8/29 -- We decided to have homeworks due on Wednesdays, so HW1
will be distributed at the next class period, to be due the following
Wednesday. Class discussion was a couple of inductions, and then some
axiomatics. The reals are a complete Archimedean ordered field. There
are still more people who want to take the course than there is room.
The departmental policy is to limit to 28 students, and I'll stick to
that. This course is offered every semester, and as of Friday at
least, there was an opening in the other section. A student suggested
an alternate way for me to make these files, so here is another copy
of the Class
Organization. Another student asked about proofs that pi and e are
transcendental (that is, not algebraic). Rather than make up a lot of
handouts, let me give you the following link
Sci.math
discussion of irrationality and transcendence of pi and e.
W 9/3 -- More on the reals as an ordered field. The important
thing to get out of this is the ability to manipulate inequalities
involving the absolute value. We began to talk about the supremum and
the infimum. We'll see much more about this later in the course.
First assignment was made, due W 9/10: Homework
One. Thanks for the improved .pdf fonts are owed to Pinaki
Chakraborty.
F 9/5 -- Information about the TA's office hours was distributed;
find it at the top of this page.
An introduction to upper bounds and lower bounds and
suprema and infima. The significance of completeness and the
Archimedean property. One small correction: after class, an alert
student observed an error in my version of the proof of Theorem 4.7 --
that the rationals are dense in the reals. I'd defined on the board
the set {j: an < j < k} and taken its minimum element without
demonstrating that it was non-empty. In point of fact, what I'd done
was misread my notes, it should include j less than or equal to k.
M 9/8 -- A handwritten handout on countability was distributed and
discussed, but it's unlinked. Come to class. We went through the
definition of the limit of a sequence, with a couple of examples.
It was pointed out after class that the book writes sequences with
parentheses (s_n) and I wrote them on the board with curly braces
{s_n}. I'll try to be consistent with the book in the future.
W 9/10. -- Homework one solutions distributed (unlinked), come to
class to get them! Second assignment was made, due W 9/17: Homework
Two. Various examples of sequences and limits and manipulations
therein.
F 9/12. -- A small mystery about enrollment. The class limit was
raised somehow.
Homework one graded, and returned, along with an
additional sheet of comments and alternate proofs. I gave two distinct
proofs for HW1,#7 in class, then continued with sequences. We will be
starting with Section 10 in class on Monday. Last five minutes of
class were about iterated square roots. We'll see more of this on
Monday.
M 9/15 -- No major questions about the homework. I proved that
if (s_n) is a convergent sequence which converges to s, and k is a
fixed positive integer, then the sequence (s_(n+k)) also converges to
s. We went through section 10 with the definitions of monotone
sequences, the basic theorems about their convergence and an
introduction to lim sup and lim inf, which we'll be seeing a lot more
of in the future. We didn't get to the definition of a Cauchy
sequence, which will be done on Wednesday. We did discuss iterated
square roots, as a pleasant example of a monotone sequence.
W 9/17 -- Completion of proof that lim inf s_n = lim sup s_n = s implies
lim s_n = s, more examples of sequences and their lim sup's and lim
inf's and definition of a Cauchy sequence.
Oh yes, HW 2 due,
handwritten solutions distributed, with an error in #10b that was
corrected on Friday's supplemental comments.and Homework
Three, due W 9/24, passed out.
F 9/19 -- Best class of the semester (from my biased point of view
at least). Homework returned, along with the Retrospective, and I
shared a memo about how easy it is to drop this class (d'oh) -- see
comments at the top of this page.
Finish proof that a sequence of reals is Cauchy if and
only if it is convergent. Start of discussion of subsequences and the
"Whitman sequence", which has subsequences converging to every real
number in [0,1]; 0, .1, .2, ... , .9, 1, 0, .01, .02, ...., .99, 1, 0,
.001, ... , .999, 1, ... Variations which give subsequences
converging to any real number. Last 15 minutes was an elaboration of
HW2 #8, on the iteration of the function f(x) = (6x - 8)^(1/2) on
[2,4] and elsewhere.
M 9/22 -- Some theorems about subsequences, with the temporary
introduction of the weak subsequence: if (s_n) is a sequence
and (m_k) is a sequence of positive integers so that m_k -> infinity,
then (s_(m_k)) is a weak subsequence. (The difference from the usual
subsequence is that (m_k) does not have to be monotone. This
simplifies some of the proofs in section 11; not enough, however, to
make them extremely interesting.
W 9/24 -- Homework 3 collected and Homework
Four, due W 10/1, passed out. Material covered was sections 11 and
12, with a proof that if s_n = n!/n^n, then s_n^(1/n) -> 1/e. I
misleadingly wrote this as s_n ~ 1/e^n, and will correct this on
Friday. Class ended with a proof of Cauchy-Schwartz.
F 9/26 -- Homework 3 returned, along with supplemental notes.
A few final remarks about section 12: mainly, that the negation of
the statement "lim s_n = s" is not the statement "lim s_n = t, where
t is not equal to s", because the limit might not exist, and the
sequence might not diverge to plus or minus infinity. We started on
section 13, with a variety of metric spaces introduced and discussed.
Monday will focus on topology.
M 9/29 -- A day of topology. We nearly, but not quite, finished
section 13. I said we'd finish it some time in the future.
W 10/1 -- Homework 4 collected and Homework
Five, due W 10/8, passed out. Despite my promise on 9/29, I
finished section 13, with the proof of the Heine-Borel covering
theorem. Introduction to infinite series.
F 10/3 -- Homework 4 returned, along with more notes. Big point:
notation is important, and it can convince you that mathematical
truths are true when they haven't been proved yet. More on series,
with the Cauchy Condensation Test (see handout Monday?). Also, a
description of the Cantor set.
M 10/6 -- We finished series and talked about the first test.
A handout on Cauchy Condensation was distributed. Since
there was no agreeable evening time, we settled on W 10/15 in class.
We did more on series and summarized section 16 as quickly as possible.
W 10/8 -- Homework 5 collected. Discussion of continuity begins; this will be
on the second test, not the first. Homework 6 will be distributed
on M 10/13.
F 10/10 -- Homework 5 returned, with a more detailed discussion of
series and a bit more on continuity. Two examples of sequences of
continuous functions were discussed.
M 10/13 -- Review for test 1. More on continuity, including the
Intermediate Value Theorem and the fact that a continuous function
achieves its maximum and minimum on a closed bounded interval.
W 10/15 -- Test 1.
F 10/17 -- Completion of the very important section 18, with a
discussion of the inverses of strictly increasing continuous functions
and a glimpse of non-intuitive continuous functions, such as those
defined by the Cantor set.
M 10/20 -- Test 1 will be returned, HW 6 will be collected and
HW 7 will be distributed. Section 19 will be covered.
W 10/22 -- Today and Friday, Prof. Tumanov will cover the class while
I am at a conference in Dortmund, Germany.
F 10/24 -- F 11/21. I kind of lost it trying to catch up after
Dortmund. But here are the links to HW's 6, 7, 8, 9, 10 and 11.
I'll put reminders at the top regarding the time and place of
test. What we covered on 11/21 was section 27 on the Weierstrass
Approximation Theorem.
Homework 9 had these corrections: in Problem #8, f(x) is the power series given; in Problem #10b, in order to answer the second part, you need a different series than the one I gave in the first part, so throw in a factor of (-1)^n. Thus you get from 10a to 10b by taking 3 to 4. The answers look more different than that, however.
Homework 10 had these corrections: #7 -- M should be sup { | f'(x) | : x in [0,2] } -- note absolute values. I'll be generous in the grading, but think of a theorem that should be applied.
#8 -- In the parenthetical remark, the second F(x) should be x/(1+x) not x^n.