@mr handy, This had EVERY chance to turn into a productive interesting discussion -.-
Just to remind you, I'm angry because you ignored & didn't respond to my posts, which I did in fact put effort into. Even if you're arguing against an army of bigoted people who will never admit you're wrong, you don't actually have to start insulting them. Open conversation is a possibility!
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Just for summary's sake, my real analysis take [ie, what I'd write if a math professor asked me to give a watertight proof]:
- The decimal number 0.(9), (denoting 0.99... repeating) represents the real number which is the infinite sum of {0.9,0.09,0.009,0.0009,...}.
- Infinite sums are defined as the limit of the sequence of partial sums.
- The sequence of partial sums is {0.9,0.99,0.999,...}
- The limit of this sequence is 1.
[
Proof that the limit of the sequence is 1: Denote L=1. The definition of a limit of a sequence A(n) is, "A(n)->L as n approaches infinity means that for any e there exists an N such that all n>N have |A(n)-L|<e". This states that A(n) gets arbitrarily close to L, for sufficiently large n.
Let e be arbitrary and let A(n) be the nth term of (0.9,0.99,...}, with A(1)=0.9. Clearly A(n)=1-10^(-n). If we choose any integer N with N>-log(e), we see that n>N has |A(n)-L|=|1-10^(-n)-1|=10^(-n)<10^(-N)<10^(-(-log(e)))=10^(log(e))=e, so that |A(n)-L|<e for sufficiently large n.
]
QED.
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I have a feeling that if you understood that proof, you'd say, "Oh, well, that's the definition I meant", to which I'd say, I'd love to see your definition, but note that that's the definition that massive,
massive fields of mathematics are based on.
I could also write a similar proof that there exists no largest number less than one. But one could say, "those properties you're using aren't properties of numbers I know of", but you have to realize that questions like these were really heavily thought out by Cauchy and Weierstrass and Bolzano and so many people, and these were the definitions they converged on, so their merit might not be easy to see.
"I <3 u 2 bbz" - Dark Frager
