Yes, there are several more or less elegant explanations for this. One I find particularly intriguing is based on the notion that on |R (real Numbers), you can always find a number between two numbers, i.e. there exists an y with x < y < z for all x, z with x not equal z. Therefore, if we cannot find an y, x equals z.
Let's just try this out and see what happens:
i) If we try to find a number between 0.999 and 1, our first idea would be to just append another 9 to 0.999, giving 0.9999, which leads us to: 0.999 < 0.9999 < 1, which is correct.
ii) We can do this ad nauseam: Appending one more 9 still fits x < y < z
iii) But 0.999... has infinitely many 9's -- we just can't append more 9's to create a number between it and 1, because 9 is the largest digit we can use in base 10, leading us to the answer to the question whether there is a y for:
0.999... < y < 1; 0.999 != 1
The answer is NO.
As we proposed earlier, this directly leads us to
0.999... = 1
There are countless other proofs, but I find this one especially intriguing, given its non-reliance on actually calculating anything.
Name:
Anonymous2008-07-31 6:38
>>7
Between Unicode ℝ and TeX \mathbb{R} you went with |R?
Though your proof is correct, of course.
Name:
Anonymous2008-07-31 8:58
>>6
If anyone's interested in that script, it's basically http://cairnarvon.rotahall.org/misc/progfind with all instances of ``prog'' replaced with ``sci''.
That way at least something useful came out of this troll.
You'll need a Perl interpreter and a command line interface. The guy who wrote it wrote it for Linux, but it should work under other operating systems too.
>>23
Is it? OR maybe it is not true. >>24
Well obviously something is wrong with the argument, because there is no final term, and n will never be infinity.
>>25
I know /sci/ is close to the lowest common denominator for maths-related boards, but I still didn't expect to find someone who doesn't understand limits.
IHBT.