@Srivatsan Looks quite fine to me; you never sounded like you were "talking down". I would add that he should see Bressoud's real analysis book as well for a nice view of epsilon-delta.
@Matt Knuth has an idiosyncratic way of writing various Greek symbols... Trouble is that the Stix version of \varrho is even uglier than \rho, for example.
@ZhenLin Oh wow. Let me see what that is. (In any case, for all my private purposes, I have defined a newcommand called \eps which is varepsilon, so I am not usually that bothered by this. =))
Heh, I was convinced commutative algebra wasn't up my alley, but I knew I would need to study it if I wanted to make any headway in algebraic geometry. So I persisted in banging my head against the wall.
@Matt Oh my. Don't remind me. Yesterday I ran into two ladies who were so excited by the comic sans-ish font one of them managed to install in her iPhone.
@JM: Admittedly, I cheated when I started learning because I could read Chinese already, but since then I've ended up having better reading comprehension for Japanese than Chinese...
I must confess, I never learned enough to read that...
I remember though, being in a calligraphy-tools shop and hearing some Japanese students try to read the samples and actually being able to make some sense out of them!
@JM Have you always been learning languages since childhood? Or did it "start" at a later point? (E.g., when you had to travel around for some reason...)
@JM Oh, I was most impressed with the Filipino, actually. =)
The funny thing is, we talk about private corporations running the world nowadays, but it was literally true back then, what with the British East India Company...
@Srivatsan Because it's quite off topic at the moment : )
Assume you have a finite open cover $B(\delta_i, x_i)$ for $0 \leq i \leq n$ and you want to pick $\delta := \frac{\min_i \delta_i}{2}$ and cover the space using $\delta$ balls, what's a good way to construct this? I'd like to write down the new cover but for that I'd need to specify the new $x_i$ I guess...
Hey guys, can someone please help me with finding a bijection between $[0, 1] \cup [2, 3] \cup [4, 5] \cup \cdots$ and $[0, 1]$? I know that they both have the same cardinality because the first one is countable union of continiums and the latter is a continium, but I struggle to find a specific bijection.
I've also read that it's possible to construct a bijection between $\mathbb{N} \times \mathbb{N}$ and $\mathbb{N}$ as a mathematical formula (it actually says here that it's a polynomial of deg = 2 with coefficients from Q) as opposed to description in English.
@Srivatsan A pair of injections would do too.
I've also read that it's possible to construct a bijection between $\mathbb{N} \times \mathbb{N}$ and $\mathbb{N}$ as a mathematical formula (it actually says here that it's a polynomial of deg = 2 with coefficients from Q) as opposed to description in English.
And I was wondering, is there any way to approximate integrals such as $$ \int_0^{\infty} f(x) dx ?$$
Mmm, perhaps if one were to take the expantion at infinity instead of at zero it could work... Mmm
For an example, how would one approximate the integral $$ \int_{0}^{\infty} \cfrac{1}{1+x^4} $$ ? I know you can solve it using countur-Integration. But lets pretend you only want a simple estimate.
Wikipedia is terrible for learning new things in mathematics, atleast to me. If I feel I know a subject to some degree, It is quite good for developing a deeper understading. But If you have absolutely no clue about a specific subject, wikipedia can be a rather tough place to start learning.
@tb Well there are different degrees of think you're due a major prize. Max Born thought for decades that he'd been cheated out of the Nobel Prize by Heisenberg and Schrödinger -- which was quite justified and didn't make him a crank. But I'm also sure that he didn't think it a viable way forward to exhort random strangers to nominate him.
@ZhenLin That is definitely something I agree with. [Unless the joke is that even such elaborate jokes could start to feel funny after some time... =)]
@Asaf: There are lots of things you could add. Like what all the fuss about the Principia Mathematica proof is about. Or why some people might not accept the Peano axioms. etc.
I think that what the 1+1=2 guy needs most may be a touchy-feely lecture on how axioms are things we choose to work with and symbols just mean what we decide they mean, etc. His main ignorance may be about what it is reasonable to expect from formal proofs in the first place.
Perhaps also note explicitly that the fact that such-and-such formal system can prove 1+1=2 is an interesting fact about the formal system but does not tell us anything interesting about 1+1=2 itself.
In fact, I'm inclined to think that bringing up the PA proof without such a qualification is a misleading approach. (This also goes for my comment to the question).
