Date: Nov 13, 2012 5:11 PM
Author: Uirgil
Subject: Re: Cantor's first proof in DETAILS

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?
>
> -LV
>


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.

Density is not enough distinguish between Q and R, but the GLB/ LUB
property is enough.

Any densely ordered interval of positive length having the GLB/LUB
property is uncountable.