```Date: Nov 13, 2012 1:58 AM
Author: Zaljohar@gmail.com
Subject: Re: Cantor's first proof in DETAILS

On Nov 12, 9:18 pm, "LudovicoVan" <ju...@diegidio.name> wrote:> "Zuhair" <zaljo...@gmail.com> wrote in message>> news:86a85cce-2a84-4c9f-b860-527958274b50@o8g2000yqh.googlegroups.com...>> > Let a_0 = x_0> > Let b_0 be the first entry in (x_n) such that b_0 > a_0.> > Let a_i+1 be the first entry in (x_n) such that a_i < a_i+1 < b_i.> > Let b_i+1 be the first entry in (x_n) such that a_i+1 < b_i+1 < b_i.>> In Cantor's proof a_{i+1} and b_{i+1} are the two first entries encountered> (in any order) in (x_n) *after* the entries corresponding to a_i and b_i.> This does not seem to be the case with your proof, where it instead seems> that entries are just picked every time restarting from the beginning of> (x_n).>> Could you clarify?  I'd like to be sure before I proceed reading it...>> -LVYes in this proof the entries are as you said will be picked everytime restarting from the beginning of (x_n) provided you follow therules stipulated in definition of the a's and b's. BUT you'll see asyou go down the proof that this will also lead to every a_i+1 and b_i+1 coming after a_i , b_i in (x_n). (See Result 6 of this proof).As regards the "order" of a_i and b_i, then if you meant by order thequantitative comparison of their "values" then clearly in this proofwe MUST have a_i < b_i by definition. But if you meant by "order" theplaces of them in (x_n), then the mere definition of the a's and b'sdo not mention itself any place order restriction, but as I said stillby following those rules this will eventually lead to each b_i comingafter a_i in (x_n) [Result 4].Also I agree with your notation that a_i+1 is better written as a_{i+1}. But I guess it is understood like that anyway.Zuhair
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