More on proofs

[ciphergoth]

Some really interesting answers on what proofs you know. The most common proofs people mention are the ones I name, plus Cantor's diagonalisation argument that |R| > |N| - cool. More specific comments follow.

wildbadger -- product of two compact spaces is compact

A strong opening! I looked it up and found Tychonoff's Theorem - looks interesting.

cryptodragon -- A number of them from crypto stuff

Interesting, name us a favourite?

simple_epiphany -- As a third-year maths student, I'm required to know quite a lot of them, but the one for the Bolzano-Weierstrass theorem is quite nice.

Bolzano–Weierstrass theorem on Wikipedia. I think I could remember that proof. Cool, thanks!

aegidian -- Cantor's Diagonalisation, proving there are as many rational numbers as there are integers.

Ah, the proof that |Q| = |N| rather than the proof that |R| > |N|?

olethros -- Maybe I lied. I can prove (by recursion) that all marbles in the world are the same colour.

I know that proof :-) Oh go on, there must be a *valid* proof you like!

keirf -- Bolzano-Weierstrass theorem - every bounded sequence in R{n} has a convergent subsequence

A second showing for this theorem!

ajva -- that 0.999...=1

Don't you need to get into the construction of the real numbers to explain this one?

ergotia -- The infinite number of primes/hotel at the end of the universe one

Two proofs for the price of one :-)

nikolasco -- irrationality of sqrt(2) (fundamental theorem of arithmetic, even/odd, well-ordered)

What proofs are you referring to with "even/odd, well-ordered"?

Thanks all, please keep commenting :-)

Comments

[emarkienna.livejournal.com]

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.