## Wednesday, July 30, 2008

### 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 $\{p_n\}$ is a Cauchy sequence in a metric space $X$, and some subsequence $\{p_{n_i}\}$ converges to a point $p\in{X}$. Prove that the full sequence $\{p_n\}$ converges to $p$.
2) Let $\{E_n\}$ be a sequence of closed, nonempty, bounded sets in a complete metric space $X$. Also $E_{n+1}\subset E_n$ and $\displaystyle\lim_{n\to\infty} diam E_n=0$. Prove that $\bigcap_{n=1}^\infty E_n$ 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.