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

[damerell.livejournal.com]

Of course the best proof of all is that e ^ (i pi) == -1 but I'd need to brush up on that one and there is too much meat on it for a party, assuming my other partygoer doesn't have a relevant background.

[makyo.livejournal.com]

You can prove e^iθ = cosθ + isinθ using Taylor–Maclaurin series (as long as you assume that the series converge everywhere in the complex plane - which I believe they do, but it's a while since I've done any complex analysis). Then set θ=π to get the special case e^iπ=-1.

[nikolasco.livejournal.com]

Wikipedia has a few proofs (http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs), including the one makyo alluded to.