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

More proofs

[(Anonymous)]

(You don't know me, but I came across this, and I can't help but reply; and hopefully with a couple interesting ones.) As someone who did a degree in maths, I could hopefully do quite a few. But the two I *like* (of the ones not previously mentioned) are:

• Construction by straightedge and compass giving algebraic extensions of degree power-of-2, and then proving the non-constructibility of certain numbers, giving you three unproven-for-thousands-of-years theorems at once: The Impossibility of Doubling the Cube (http://en.wikipedia.org/wiki/Doubling_the_cube), Trisecting the Angle (http://en.wikipedia.org/wiki/Trisecting_the_angle), and Squaring the Circle (http://en.wikipedia.org/wiki/Squaring_the_circle). (More (http://en.wikipedia.org/wiki/Constructible_number))
  
  
• Not-Burnside's Lemma (http://en.wikipedia.org/wiki/Burnside's_lemma). I don't know why, but I love this theorem. (And it's got the interesting weirdness with its name. See the "History" section.)
  

Re: More proofs

[ciphergoth.livejournal.com]

Funnily enough, I was looking at that the other day - in fact, you could check if the bit I added that says "the constructible numbers form the smallest field extension of the rational numbers which is closed under square root and complex conjugation" is correct!

BTW, anonymous posts are welcome but please sign with a nym!!

Re: More proofs

[insipidia.myopenid.com (from livejournal.com)]

To be honest, I can't really check that, since I haven't done any abstract algebra in 6 or 7 years. :) But it does *sound* right. (If it's clear to the reader that when you say "smallest", you mean "any other such field contains this one".) I can't think of a counterexample, anyway; although I am really rusty.

(Oh, and sorry about the anonymity. I don't use my LJ account any more, and I always forget that LJ accepts OpenID now. Old habits die hard. :) )
(Oh, and also sorry about that first badly formatted comment. I thought links were preserved properly.)