Thursday, October 01, 2009

The Bolzano-Weierstrass Theorem

Don't worry - I'm not going to try to explain the Bolzano-Weierstrass Theorem. I just like the name. Don't you?


We use it a lot. There is something really fun about using a theorem with a name like that, and it only improves with repetition.


Advanced Calculus was really hard last night. What's happening these days is that we suddenly have a whole bunch of theorems, each of which relies on the previous theorem, going back to the aforementioned Bolzano-Weierstrass. (Of course, Bolzano-Weierstrass has its own antecedents, but I understand them already.) The last in this series is the Intermediate Value Theorem, which I think is the first time we've had a theorem I recognize the name of from calculus. So, we're finally up to calculus. Epsilons and deltas roam freely across the land.

Yes, I am a bit loopy with exhaustion today. Why do you ask?

Anyway, what's challenging about doing a bunch of theorems in a row is that, when you barely understand one theorem, using it to prove another theorem leads to confusion. And when you do five or six theorems in a row that way it starts to suck, and you sit like a zombie in class, mechanically copying from the board.

Hopefully tonight or this weekend I can sit down and go through all of the theorems, starting with Bolzano-Weierstrass, and their proofs. By "go through" I mean "write down from scratch, making sure that I understand what the theorem is saying and every step in its proof." I think this has to take priority over the homework due next Wednesday for the sake of my understanding of the material yet to come.

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