Well, that was an interesting experience. It's taken me more or less all day, but I've gotten through a couple of problems in Stichartz's real analysis book. It didn't take me so long not because Strichartz's book is spectacularly difficult, which I suspect it isn't, for people who a) have some aptitude for math and b) are using it in a class, and can therefore seek outside aid if some part of it just doesn't make sense despite vigorous squinting. But what with my world-renowned math skillz and isolation, it can be fairly harrowing.

Now, this isn't the first time it's taken me obscenely long to solve some exercizes out of a book. What was interesting here was that it was intuitively obvious what the solutions were after a few minutes of playing with each of the problems. For instance, there was one about proving that given any real number x there's a cauchy sequence of rationals converging to x where all the terms of the sequence are less than x. Now, it's *obvious* that there have to be cauchy sequences like that - they're the ones that approach x sedately from below, rather than jumping all around it first, or falling on it from above. I even came up with a construction for an example of such a sequence - the first term is just some rational number less than x, and then x_j+1 is given by any rational number such that x_(n+1) > (x + x_n)/2 (and the idea is that x_n < x_(n+1) < x), which we can always get based on the density of the rationals in the reals. The idea is that each subsequent term more than halves the distance to x, so that the whole shebang gets arbitrarily close to x as you crank n up to as high as you want. For some reason, formally arguing that sequences defined like above are cauchy sequences, and that they are cauchy sequences of x, was really, really hard for me. That seems strange, to put it mildly - but that's in hindsight.

Anyway, the benefits of going through all this just to prove something obvious are that I get some experience writing proofs, and that I've really memorized the definition of a Cauchy sequence now. If you wake me up in the middle of the night, and ask me "What's a Cauchy sequence of rational numbers?", I'll probably be able to answer (though not without some frothing at the mouth). ("It is a sequence of rational numbers {x_k} which 'come together' - meaning that given a natural number N, it is possible to find an m, depending on N, such that for all h,j ≥ m, | x_h - x_j | ≤ 1/N - meaning that given past a certain point in the sequence, all subsequent terms of the sequence are no more than 1/N from each other.")

## 0 Comments:

Post a Comment

<< Home