LudovicoVan
Posts:
2,971
From:
London
Registered:
2/8/08
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Re: Cantor's first proof in DETAILS
Posted:
Nov 13, 2012 5:20 PM
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"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
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