Virgil
Posts:
4,491
Registered:
1/6/11
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Re: Matheology § 162
Posted:
Nov 27, 2012 2:28 PM
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In article <ahkgehFjgu0U1@mid.uni-berlin.de>,
Carsten Schultz <schultz@zedat.fu-berlin.de> wrote:
> On 27.11.12 19:06, WM wrote:
> > On 27 Nov., 14:15, Carsten Schultz <schu...@zedat.fu-berlin.de> wrote:
> >
> > The sequence (a_n) with
> > a_n = (((â¦((((((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.
>
> Oops, wrong example. What you complain about is that for
>
> b(k,n) = 10^{k-n}, if n<=k<=2n, k odd,
> b(k,n) = 0, otherwise
>
> we have lim_{n->oo} sum_{k=0}^oo b(k,n)=oo even though
> lim_{k->oo} b(k,n) = 0 for all n (and this is a limit of an eventually
> constant sequence). Surely set theory must be to blame for this!
Of course!!!
WM is in the habit of blaming set theory for everything.
>
> > 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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