Two weeks in the books

Two weeks of UPenn’s math camp are in the books!

We’ve covered a lot of material in the last two weeks, including:

Dedekind cuts, Cauchy Sequences, Cardinality ,Continuity, Metric Spaces, Norms, Inner Products, Boundedness, Open and Closed Sets, Completeness, Compactness, Coverings, Connectedness, Clusters, and Correspondences.

This material is NOT EASY. There isn’t always time for me to fully understand each concept before we have to move on to the next one. In particular, I have difficulty coming up with creative ways of defining sequences and epsilon-delta conditions. If I can solve a problem, it’s usually because I can think through each step logically and convince myself that it must be true. But, I need more practice formulating my logical arguments using specific mathematical language. At the moment, my proofs lack rigor because I don’t define the sets exactly right or I don’t specify the right radius for my open epsilon-ball. I’m hoping that stuff will come with time.

There are close to 40 students in math camp between the UPenn Economics, Wharton Finance, and Wharton Applied Economics programs. All but three of them have taken a course in Real Analysis before (I’m one of the three!), and many have taken courses in Topology. About half of the cohort has a serious math background, by which I mean that they have at least an undergraduate degree in math, and many have graduate degrees in math or statistics.

This all means that I’m not as well prepared for math camp as many of my colleagues. However, I benefited enormously from reading Velleman’s How to Prove It before I got here. From the very first day of class, it was clear that reading the book (and working through the exercises) had greatly enhanced my ability to comprehend the structure of the proofs I was reading, and break down my own proofs into intermediate goals. The book also helped by giving me plenty of practice with the contradiction and induction proofs which form the meat and potatoes of our course thus far.

I actually can’t recommend How to Prove It enough. I think it should form the basis of a required course for first-year college students in all majors. It teaches you how to logically order arguments. People who go on to do a major that requires a lot of writing would probably benefit indirectly from the training that the book provides. How to Prove It is especially valuable, though, if you plan on doing some graduate work involving mathematics.

That’s all for now. I’ve got to get back to studying