Here are the pictures from today's class.
Here is the syllabus.
Here are the notes.
Here is the facebook page.
Presentations: Katie gives a nice alternative proof to T24. Aaron makes a good start on T25.
Presentations: John T11, Ted does there exist a closed, perfect set containing no interval? Ted - integration.
Presentations: Tre P30, Kim T18.
Tre there is a link to the facebook page above. You made good progres on P30 today and I think that it will
be beneficial to you to write up the argument that you gave. Others in the class might like to see a write up and if you will turn a write-up in to me, I will
help you make sure that it is complete to help you with future problems.
Presentations: Lucas T27, Ted, brief discussion on P31.
Welcome to my Analysis II blog. We also have a Facebook Group, Analysis4326.
If you request to join it, I'll let you in. Don't worry, we don't have to be facebook "friends"
in order to both be members of the group. What you post there will only be visible to our class.
We'll start out where we left off last semester. Outstanding Problems: T17, P27, P29,...
Presentations: Jessica T23, Tre P30 (good start!), Lucas T24 ("professional" in the words of Tre!).
Welcome to my Analysis II blog. We also have a Facebook Group, Analysis4326.
If you request to join it, I'll let you in. Don't worry, we don't have to be facebook "friends"
in order to both be members of the group. What you post there will only be visible to our class.
We'll start out where we left off last semester. Outstanding Problems: T17, P27, P29,...
Presentations: Rabbi T25, Lucas T16
Here are the pictures from today's class.
Presentations: Amber T21, Lucas P22
Exam Friday at 8 - 10:30 am. One problem on language (definitions/negation),
one problem on True/False (prove
or give counterexample), several proofs where you may choose to omit one of them.
Here are the pictures from today's class.
Presentations: Kim T20, Milagro T15
Here are the pictures from today's class.
Presentations: Weston - Product of Continuous Functions -- nice.
Here are the pictures from today's class.
Presentations: Clint T21, Shamal P28
Here are the pictures from today's class.
Thanksgiving!
Presentations: Clint T21, Shamal P28
Here are the pictures from today's class.
Presentations: Han T18, Jessica T19
Here are the pictures from today's class.
Presentations: John T11, Kim T20
Here are the pictures from today's class.
Presentations: T14 Lucas, Differentiability - Ted
Here are the pictures from today's class.
Presentations: Anthony T13, Kim T20
Pix
People I advise keep sounding worried about their grades or about "surviving" this class.
There are no individuals who are regularly attending about whom I have concerns. Grades on
the final are always better than the midterm because you have 2.5 times as much time and the
exam is only slightly longer. Also, you have plenty of time left to present and you are all
becoming better at proving and writing proofs.
Presentations: Lucas P22, Milagro T12, Ted - 30 second intro to sequence version of continuity
Don't forget to let me know if
1. you are interested in a one hour 4131 based on a continuation
of 3380(now 4325).
2. you are interested in 4326, but have a schedule conflict.
Recall that the only difference between 3380(4325) and 3381(4326) is
that we have no tests and no written homework, we just talk about problems in class everyday
until we all agree we have a proof!
Here are the pictures from today's class. Weston, I apologize that I don't have pictures of
your work. I'll happily reproduce it tomorrow if anyone would like to see it.
People I advise keep sounding worried about their grades or about "surviving" this class.
There are no individuals who are regularly attending about whom I have concerns. Grades on
the final are always better than the midterm because you have 2.5 times as much time and the
exam is only slightly longer. Also, you have plenty of time left to present and you are all
becoming better at proving and writing proofs.
Presentations: Michael P23, Weston P26
Don't forget to let me know if
1. you are interested in a one hour 4131 based on a continuation
of 3380(now 4325).
2. you are interested in 4326, but have a schedule conflict.
Recall that the only difference between 3380(4325) and 3381(4326) is
that we have no tests and no written homework, we just talk about problems in class everyday
until we all agree we have a proof!
Here are the pictures from today's class. Weston, I apologize that I don't have pictures of
your work. I'll happily reproduce it tomorrow if anyone would like to see it.
Presentations: Lucas 22 (more soon?), Rabbi 25, Ted intro to T15,T16
Don't forget to let me know if
1. you are interested in a one hour 4131 based on a continuation
of 3380(now 4325).
2. you are interested in 4326, but have a schedule conflict.
Recall that the only difference between 3380(4325) and 3381(4326) is
that we have no tests and no written homework, we just talk about problems in class everyday
until we all agree we have a proof!
Here are the pictures from today's class.
Presentations: Milagro T4, Ted - "relating limits to our definitions"
Don't forget to let me know if
1. you are interested in a one hour 4131 based on a continuation
of 3380(now 4325).
2. you are interested in 4326, but have a schedule conflict.
Recall that the only difference between 3380(4325) and 3381(4326) is
that we have no tests and no written homework, we just talk about problems in class everyday
until we all agree we have a proof!
Here are the pictures from today's class.
Presentations: Clint W. T10, John T11 (more on Friday?)
Here are the pictures from today's class.
Happy Haloween!
Presentations: Katie -- Lemma to 17, Ted -- T17 revisited, Anthony P21
Here are the pictures from today's class.
Presentations: Aaron T7, Clint P20
Clint G., Please accept my apologies for not photographing your nice work! I'm terribly
sorry that I didn't get a record of P20. Ted :(
Here are the pictures from today's class.
Presentations: Aaron T7, John T11
Here are the pictures from today's class.
Presentations: Jacob T8, Chris P24
Here are the pictures from today's class.
Presentations: Han T9, Weston P25
Midterm: Wednesday. Glad you didn't vote for Friday or you'd have wanted the weekly write-up off. ;>
Here are the pictures from today's class.
Midterm
Presentations: Anthony T4 (more on Wednesday?), Amber T5, Ted -- midterm, completeness, nested intervals
Midterm: Wednesday. Glad you didn't vote for Friday or you'd have wanted the weekly write-up off. ;>
Here are the pictures from today's class.
Presentations: Anthony T4 (more on Monday?), Aaron T6 -- it's nice to see good mathematics
going on the board every day now -- even when not complete, you are writing well.
Midterm: Next Wednesday. Glad you didn't vote for Friday or you'd have wanted the weekly
write-up off. ;>
Here are the pictures from today's class.
Here is the quote by Edwin E. Moise that I read in class:
"I believe Moore’s work proves something of broad significance... that sheer knowledge
does not play the crucial role in mathematical development that most people suppose.
The amount of knowledge that a small class can acquire, struggling at every stage to
produce its own proofs, is quite small. The resulting ignorance ought to be a hopeless
handicap but it isn’t. The only way I can see to resolve this paradox is to conclude
that mathematics is capable of being learned as an activity and that knowledge which
is acquired in this way has a power which is out of all proportion to its quantity."
Presentations: Clint W. T1, Milagro T3 -- two very nice presentations! :)
Midterm Coming Soon to a Classroom Near You Next Week: Let me know when you want it or
I'll make it a POP mid-term and surprise you!
Here are the pictures from today's class.
Presentations: Amber P15, Jacob P17 -- two very nice presentations! :)
Midterm Coming Soon to a Classroom Near You Next Week: Let me know when you want it.
Here are the pictures from today's class.
Presentations: Ted -- continuity, Jessica P12
Hey, you got me to lecture and all you had to do was ask a question. Amazing. It's almost
as if I'll do anything, even answer questions, to help you understand. ;>
Today I asked a student why s/he had not presented yet and s/he said, "because these
problems are HARD." Well, true enough. If you work on one and don't have success,
COME SEE ME!
No pictures today, I forgot my camera. Too bad since Jessica put up such a nice
argument that showed that every finite set has a left most point and a right most point.
Note that this is equivalent to the statement that every finite set can be ordered from
least to greatest. The same proof works -- order a set of one element. Suppose you
can order a set of n elements. Pick a set with n+1 elements, take a point out of that
set, order it by the inductive hypothesis, then add the point back in. No matter where
it lands, the set is now ordered.
Presentations: Jessica P12, Lucas P16, Jacob P15 Friday? Katie P17 Friday?
Midterm Coming Soon to a Classroom Near You: Not this week or next -- let me know when
you want it.
Sometimes students misinterpret my stern face in class -- my face is not stern, it is
contemplative -- when in class, I am always thinking very hard either about what you are trying
to ask or explain, or I am trying to choose my words carefully so as to be both accurate and
helpful to you in my statements and questions. I am quite pleased with the work you are
doing -- all I want from you is for each of you to dig deeply into one or more problems and to
come see me if you are not making progress.
Here are the pictures from today's class.
Presentations: Jessica P12 (more Wednesday?), Jacob P15 Wednesday? Katie P17 Wednesday?
Barron -- once you think about what was discussed in class, if you have another question about the sets,
feel free to drop by the office or ask at the beginning of class.
Here are the pictures from today's class, including my comments on open, closed and continuity.
Presentations: Amber P12, Jacob P15
Here are the pictures from today's class.
Two problems! :)
Presentations: Rabbi P9, Amber P12 (more on Friday!).
Here are the pictures from today's class.
I hope that today really set the stage for success on these convergence problems! Next time
P9?, P12?, P15, P16, ... I think that it is safe to say that those who have stopped by my
office or asked me questions (about half the class so far) have walked away with a better
understanding of the problem they were working on. If you feel you are struggling or have
not made it to the board yet, I encourage you to come by with a problem that you have spent
some time on.
Presentations: Han P13.
Here are the pictures from today's class.
Great class today! You have wings now and are starting to fly! Too bad Kim and Clint G. missed
it! Jacob -- you keep questioning things, but make sure you write down a proof that
the set of reciprocals of natural numbers does not have a limit point.
Presentations: John P19, Jessica P18, Han P13 (start it on Monday before class).
Here are the pictures from today's class.
Hey, I was just looking at the photos -- how come the stuff I wrote looks so much worse
than what you all write??? ;>
Seriously, I think Kimberly did a great job today of getting us a better understanding of the relationship
between limit points and convergence. Ya'll need to come to class with good starts like she
had -- something written down that you believe constitutes a proof or an approximation of a
proof is fine. The alternative to regular volunteers from a large percentage of the class
is assigned problems to present on assigned days. You will NEVER lose any points by going to the board
and students with least presentations will be given first shot at presenting their ideas.
Presentations: Kimberly P14.
Here are the pictures from today's class.
The written work really is looking better! Keep up the good work!
Presentations: Ted (convergence, P16), Shaymal P11.
Here are the pictures from today's class.
Presentations: Ted (rehash of P8), Barron P9, discussion of alternative ways to prove P9.
Here are the pictures from today's class.
Presentations: Lucas P6, Lucas/Amber P8, Ted/Barron P9
Here are the pictures from today's class.
This was a very good day -- even though no problem was solved, we are getting the language
and the writing on the board with more precision. And I am confident that we have a good
shot at solving all three of these tomorrow!
Presentations: Aaron P6, Lucas P8 (good save), Han P9
Here are the pictures from today's class.
Math Club and Pi Mu Epsilon Meeting Friday L113 at 1:15 -- free pizza and drinks.
Presentations: Weston/Kimberly P7, Shaymal/Amber/Anthony/Lucas P10
Here are the pictures from today's class. This was a good class - lots of good ideas,
two good problems.
Math Club and Pi Mu Epsilon Meeting Friday L113 at 1:15 -- free pizza and drinks.
Presentations: Aaron P6, Barron P15, Ted P1.
Here are the pictures from today's class.
Keep up the hard work over the weekend -- they don't call it Labor Day for nothing -- you are
supposed to labor over Analysis!
Presentations: Weston and Amber P7, Shaymal P10 -- all good attempts!
Here are the pictures from today's class.
Oops! In my haste to write up what Weston needed to show if he was to find a counter-example,
I made a mistatement which is now recorded in the pictures below. :( To create a counter
example for problem 7, one would need to define a set H, define a set K, and show that
there is a point p that is a limit point of H intersect K, but is not a limit point of
ONE of the sets H and K. (I had said that it would need to not be a limit point of EITHER
set....). As I said, I won't try to fool you in these problems. To tackle problem 7, assume
that H and K are point sets (which you don't know anything about) and assume p is a limit
point of H intersect K (which must be non-empty). Now try to show that p is a limit point of
H and p is a limit point of K.
The "first point to the left" definition relates to both Problem 6 and Problem 10. Problem 10
will definitely give us a better understanding of limit points!
Presentations: Jessica P4, Lucas P5, Weston P7
Here are the pictures from today's class.
Three correct efficient presentations -- nice patch on the fly, Lucas. As a rule when one
gets stuck at the board I don't want to torture you, so you can always say "I'd like to think
about it some more" and take a seat or try to work it out on the fly.
Presentations: Barron P1, Lucas P2, Milagro P3
Here are the pictures from today's class.
Presentations: Amber Q3 (do we need to talk about that one again?); Barron P1.
Monday we'll start up with Problem 1. Here are the pictures from today's class.
A very good start. Keep up the good work.
First thing Friday I'll collect written homework. You are to write up, using your best mathematics,
one problem that has been presented in class. Because you are trying to develop your ability to
do and write mathematics, you should choose the hardest problem you think you can do well on. Remember
if you get a "C" or worse on it, you'll get a resubmit chance next week.
Friday I hope that Barron will complete Problem 1 and I'll ask if anyone can conclude Q3 by showing Amber's Conjecture that if 0<=p<=1 then
p is a lp of [0,1).
Pictures from today's class.
Please read the syllabus first.
Now read the introduction to the students in the notes.
Fill out the student interview questions and turn them in Wednesday at the beginning of class.
Now start working on the problems. Work on Questions 1, 2 and 3 from Monday's class and on Problems 1, 2, 3, ... from the notes.
Here is a picture of the class so you can get to know each other's names and a picture of the first three questions.
Click on a picture to enlarge it.
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