Date: Nov 13, 2012 5:20 PM
Author: LudovicoVan
Subject: Re: Cantor's first proof in DETAILS
"Uirgil" <uirgil@uirgil.ur> wrote in message

news:uirgil-B3AA26.15111513112012@BIGNEWS.USENETMONSTER.COM...

> In article <k7udtq$np6$1@dont-email.me>,

> "LudovicoVan" <julio@diegidio.name> wrote:

>> "Uirgil" <uirgil@uirgil.ur> wrote in message

>> news:uirgil-91F13B.13165013112012@BIGNEWS.USENETMONSTER.COM...

>>

>> > No values which are bounded below by a strictly increasing sequence and

>> > bounded above by a strictly decreasing sequence are members of either

>> > seequence.

>> >

>> > Thus proving that, given any sequence of values in R, there must be

>> > values in R not appearing in that sequence.

>>

>> I'll have a look at Zuhair's follow-up as soon as I manage, but let me

>> for

>> now just point out that the above argument is obviously bogus: the

>> rationals

>> too are dense (have the IVP as Zuhair has called it) and, by the very

>> same

>> argument, we have proved that the rationals too are not countable... see?

>

> The difference being that a monotone but finitely bounded sequence of

> rationals need not have a limit among the rationals but MUST have a

> limit among the reals, a LUB or GLB.

Yes, it's the *completeness* property that is required. Anyway, as

anticipated, I'll have to come back to this when I have time: the devil is

in the details!

-LV