Extracurricular Affairs

Well by this time I am sure that just about everyone has heard about the John Edwards affair. If not, or you for some reason want to read more about it, just click on the previous link and it will take you to an article in the New York Times. Now I am not supporting this kind of activity but really why would you put something like this out when your family is in the public eye? To me it just seems like something that you would want to keep between yourself and your spouse. Regardless, what was he thinking? He is married and even has children. Luckily his two youngest children are not that old, so the kids they will interact with most of their lives will not remember this, but it will still be one of those "family secrets" you are ashamed of. As for his oldest child (who is at Harvard Law) good luck to her and dealing with this, since you know people will want to talk to her about it and how she feels.

Also this not only hurts him and his family but also a certain democratic candidate who happens to be running for president. It also doesn't help that Edwards chose to endorse Obama and not Hillary Clinton. I am sure that the McCain campaign will you this to their advantage when the time is right.

In any case it seems that democrats seem to take part in "extramarital activities", weather holding office or not. Kennedy, Clinton, and now Edwards, just to name a few. Let's see if (and hopefully not) Obama will keep the tradition good.

solution

This is the solution for problem #1 in my previous post.

Let , since is a Cauchy sequence there exist an such that if if Similarly since a subsequence converges to we have that when . Now let be the larger of and .

Now .

Here is the solution for #2 in my previous post (this one took a little longer for me to think of).

Let be a sequence in such that . Now from properties of we have that for all Now let , and since we have that for large enough. So since is complete. Additionally, is a limit point of each , so .

From the Integers to Infinite Series

In the upcoming fall semester I will be taking the first year graduate analysis class. Hopefully, this class will be more "rewarding" than the undergraduate analysis sequence I completed last year. The undergraduate sequence was not a total waste of time since I actually did learn some new things, but the class moved so slowly and the test were not all that difficult. I never really had to exert myself in order to obtain my A. As a result in the second semester I tried to get a 100% on every assignment and test. I almost achieved this goal, however, I did not make a small clarification on one of my proof thus resulting in me losing 1 or 2 point for the problem.

The graduate class will be using the same book as the undergraduate sequence. However, we will hopefully cover all of the sections. Last year the professor would pick and choose which topics he though were necessary and possible for the majority of the class to understand. As a result some of our proofs were longer than needed but I suppose knowing multiple ways to prove something isn't all that bad.

Right now I am reading through as many chapters as I can and working all the exercises. My only requirement is that I can only be working on problems from at most two different chapters and that I can only be reading one chapter ahead of the problems I am solving. So basically right now I am working problems from chapter 2 and 3, and reading chapter 4. I am almost done with the problems from chapter 3, but the ones from chapter 2 are going to take a little more time. I suppose that it has to do with the fact that dealing with topology is still a little "new" to me.

Tomorrow I will hopefully be able to do the following problems:

1) Suppose is a Cauchy sequence in a metric space , and some subsequence converges to a point . Prove that the full sequence converges to .
2) Let be a sequence of closed, nonempty, bounded sets in a complete metric space . Also and . Prove that contains exactly one point.

I will eventually get around to making a post about my REU in Michigan. Sorry to disappoint, but I just need some more time to better collect my thought on this topic so that I will have a post worth reading.

GIMP

Well in my free time I have decided to learn how to use GIMP. For those of you that don't know, GIMP stands for GUI Image Manipulation Program. Basically it is an open source version of Photoshop.
Yesterday I started reading the online manual (which I should probably download to my computer sometime soon) and was somewhat insulted by the simplicity of the first few sections. However, I can see why they are needed. As of now most of what I know to do was discovered by playing around with the program and reading random portions of the manual.
Right now my goal is to be able to create a cool banner for the portion of this blog with the title. I already have a few done but none of them are "amazing" so they are not going to be used. I think the first thing I should find out is the necessary dimensions for the banner.
The program seems pretty straight forward so far, however, I would like to learn how to make manipulations to text.

Short Computer Post

I don't really feel like making a complete post at this point in time so I will post a a few pictures of the new desktop.


This is a shot of it in use. Running programs include: Windows Media Player, Firefox, GoogleTalk, Google Desktop, Trillian, and ObjectDock.



This is a screen shot with all windows minimized or closed.



Finally, in case you were wondering I am running Windows Vista Home Premium. There is no way in hell I would EVER get a Mac or have their OS running on my computer. As a matter of fact I try to avoid all apple products, but sadly I have to use Quicktime Player every no and again. Anyway, I don't know when but I will be trying to dual boot Windows and Luinux sometime in the future.

A New Take on Purity

Best xkcd comic strip ever.

REU 2008 - Week 1

Don't expect an entry for every week of the REU but I figured that the first week was important enough to have its own.

When I first moved into my apartment I was pleasantly surprised by the size. My first impression was that it was way too small for four people to live comfortably. However, after spending some time here I quickly realized that I was mistaken. The roommates are pretty cool, well actually I have only met 2 of them, the last one will be arriving on Monday. The first one I met had just recently returned from Africa where he was helping install water filtration systems in a village. I don't remember the name of the country but it was a French speaking one. The other roommate is a year younger than me and it kind of shows in ways, mainly when we are working in groups (he is a member of my group in addition to being my roommate). The three of us that live here so far are all math majors and the one remaining roommate is an engineering major.

The research is going extremely well. At the beginning we would "play around" with the problems and find a solution. Then our mentors (I can't think of a better term for them, I guess advisers works but that makes it seem like we are PhD students) would present us with a paper which contained the result we had just discovered. You would think that this was disheartening but it actually was not. Having actually come to the conclusion ourselves made reading some of the proofs much easier since we had a better understanding of what they were doing and could relate some of their ideas to our own. However, we have so far have come across a result which we have not been able to find stated in any paper. I guess you could say we have our first theorem, and it's quite a wonderful feeling. We were surprised by the theorem actually. This is because based on the question we were answering one would expect the result to be complicated. However, it is quite the opposite. When we first saw it we were all skeptical and furiously searched for a counterexample but could not find one. After going through it again we were certain that it was correct. All that is left to do for this finding is to actually write up the proof of the theorem and the necessary lemmas.

Well I'll keep this one relatively short and end by saying that it seems like I will be going to the beach quite often (even though we have to climb over a "mountain" to get there). So far this week I have gone three different times.