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Re: Matheology § 162
Posted:
Nov 27, 2012 1:06 PM
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On 27 Nov., 14:15, Carsten Schultz <schu...@zedat.fu-berlin.de> wrote:
The sequence (a_n) with
a_n = (((?((((((10^0)/10)+10^1)/10)+10^2)/10)+? )+10^n)/10)
has the (improper) limit infinity.
Here we have an improper limit that, according to analysis, has
infinitely many digits 1 left of the decimal point (i.e., a non empty
set), and according to set theory the same limit has an empty set of
digits left of the decimal point.
>
> Mückenheim is worried by the fact that for a sequence (a_n)_n of
> functions a_n: Z -> {0,1} it is possible that lim_{n->oo} a_n(k)=0 for
> all k while the sequence sum_{k in Z} a_n(k) * 10^k tends to infinity
> for n->oo. And of course he thinks that this is somehow set theory's
> fault. What idiocy!-
Sorry, you are plainly wrong. Your well-known text book example does
not yield a contradiction between analysis and set theory. The digits
remain left of the decimal point (if there is any point at all). Only
the positions of the digits =/= 0 cannot be determined in the limit.
This is the same in analysis and set theory. And it is obviously not
under discussion here.
In order to teach you the correct argument I have explained it again
above. Every person equipped with a minimum of intelligence should be
able to understand it after the second explanation. If you have not
yet understood it, feel free to ask again.
Regards, WM
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