Your post left me thinking how it's been years since I had to rigorously prove anything, and how much of that I've forgotten off by heart.
Cantor's diagonal proof is the first one I remember coming across that both showed a very amazing result, and seemed a clever yet simple way to prove it.
I thought of 0.9recurring = 1 (the one that involves showing that the limit of 1/10^n is 0). That's certainly a classic for Internet forums to draw out the people who think they are experts at maths, but refuse to accept any proof you throw at them (Wikipedia has a whole sub-page dedicated to it, to stop people cluttering up the talk page...) But then I realised I was struggling to remember how to do all of it.
Bolzano-Weierstrass theorem is one of the few proofs I remember the name of (I remember my tutor's advice for exams was "It doesn't matter if you can't remember the name of the theorem you are going to use, just write 'By a theorem ...'"), but I'm long past remembering the proof. Or how to spell it.
I also liked the proof of the mean value theorem, as it seems like you're just proving the bleeding obvious - but then from that it's less work to get to proving l'Hopital's rule, which isn't at all obvious and is very useful indeed. Well, I liked the idea, but ISTR I hated actually having to prove it.
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timeasmymeasure
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Date: 2009-02-20 12:11 am (UTC)Cantor's diagonal proof is the first one I remember coming across that both showed a very amazing result, and seemed a clever yet simple way to prove it.
I thought of 0.9recurring = 1 (the one that involves showing that the limit of 1/10^n is 0). That's certainly a classic for Internet forums to draw out the people who think they are experts at maths, but refuse to accept any proof you throw at them (Wikipedia has a whole sub-page dedicated to it, to stop people cluttering up the talk page...) But then I realised I was struggling to remember how to do all of it.
Bolzano-Weierstrass theorem is one of the few proofs I remember the name of (I remember my tutor's advice for exams was "It doesn't matter if you can't remember the name of the theorem you are going to use, just write 'By a theorem ...'"), but I'm long past remembering the proof. Or how to spell it.
I also liked the proof of the mean value theorem, as it seems like you're just proving the bleeding obvious - but then from that it's less work to get to proving l'Hopital's rule, which isn't at all obvious and is very useful indeed. Well, I liked the idea, but ISTR I hated actually having to prove it.