```Date: Nov 12, 2012 4:50 AM
Author: Zaljohar@gmail.com
Subject: Re: Cantor's first proof in DETAILS

On Nov 12, 12:05 pm, Zuhair <zaljo...@gmail.com> wrote:> Apologies beforehand for this long proof, and for any possible errors,> typos, mistakes that most possibly would be there with such a long> draft. I'v written this with the intention to give what I think it to> be the complete story of Cantor's first proof. So the following is my> view of this proof, it came from reading on-line proofs other than the> original one, since I don't have the original article of Cantor.> References given below.>> If a mistake in this proof is noticed, then please feel free to> outline it.>> CANTORS FIRST PROOF OF UNCOUNTABILITY OF REALS> --------------------------------------------------------------------------- ----->> Statement: There is no bijection between the set N of all naturals> and the set R of all reals.>> Proof:> We prove that for every injection (x_n) from N to R, there> exist a real J such that J not in the range of (x_n).>> Notation: for every x_i, i shall be called the place of x_i in (x_n),> while x is the value of x_i. Whenever mentioned in this article> symbols < , > , = and =/= are comparisons of the values of entries of> sequences mentioned, while the places of those entries shall be> compared by "lies before" , "lies after" , is the first entry, is the> last entry, in the same place, etc..>> (x_n) is said to have the Intermediate Value property (IVP) iff> for every two entries x_i,x_j of (x_n) there exist an entry x_k> of (x_n) such that: x_i < x_k < x_j or x_i > x_k > x_jA better statement would be:for every two entries x_i, x_j of (x_n) where  x_i < x_jthere exist an entry x_k of (x_n) such that: x_i < x_k < x_j.Zuhair
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