## Archive for January, 2014

### Introduction to Cambridge IA Analysis I 2014

January 11, 2014

This term I shall be giving Cambridge’s course Analysis I, a standard first course in analysis, covering convergence, infinite sums, continuity, differentiation and integration. This post is aimed at people attending that course. I plan to write a few posts as I go along, in which I will attempt to provide further explanations of the new concepts that will be covered, as well as giving advice about how to solve routine problems in the area. (This advice will be heavily influenced by my experience in attempting to teach a computer, about which I have reported elsewhere on this blog.)

I cannot promise to follow the amazing example of Vicky Neale, my predecessor on this course, who posted after every single lecture. However, her posts are still available online, so in some ways you are better off than the people who took Analysis I last year, since you will have her posts as well as mine. (I am making the assumption here that my posts will not contribute negatively to your understanding — I hope that proves to be correct.) Having said that, I probably won’t cover exactly the same material in each lecture as she did, so the correspondence between my lectures and her posts won’t be as good as the correspondence between her lectures and her posts. Nevertheless, I strongly recommend you look at her posts and see whether you find them helpful.

You will find this course much easier to understand if you are comfortable with basic logic. In particular, you should be clear about what “implies” means and should not be afraid of the quantifiers $\exists$ and $\forall$. You may find a series of posts I wrote a couple of years ago helpful, and in particular the ones where I wrote about logic (NB, as with Vicky Neale’s posts above, they appear in reverse order). I also have a few old posts that are directly relevant to the Analysis I course (since they are old posts you may have to click on “older entries” a couple of times to reach them), but they are detailed discussions of Tripos questions rather than accompaniments to lectures. You may find them useful in the summer, and you may even be curious to have a quick look at them straight away, but for now your job is to learn mathematics rather than trying to get good at one particular style of exam, so I would not recommend devoting much time to them yet.
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When I got to my office, those other things I’ve been thinking about (the project with Mohan Ganesalingam on theorem proving) commanded my attention and the post didn’t get written. And then in the evening, with impeccable timing, Pavel Pudlak sent me an email with an observation that shows that one of the statements that I was hoping was false is in fact true: every subset of $\{0,1\}^n$ can be Ramsey lifted to a very simple subset of a not much larger set. (If you have forgotten these definitions, or never read them in the first place, I’ll recap them in a moment.)