Date: Nov 13, 2012 5:20 PM
Subject: Re: Cantor's first proof in DETAILS
"Uirgil" <firstname.lastname@example.org> wrote in message
> In article <email@example.com>,
> "LudovicoVan" <firstname.lastname@example.org> wrote:
>> "Uirgil" <email@example.com> wrote in message
>> > 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
>> now just point out that the above argument is obviously bogus: the
>> too are dense (have the IVP as Zuhair has called it) and, by the very
>> 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!