```Date: Nov 27, 2012 1:52 PM
Author: Carsten Schultz
Subject: Re: Matheology § 162

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 = (((?((((((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,           otherwisewe have  lim_{n->oo} sum_{k=0}^oo b(k,n)=oo  even thoughlim_{k->oo} b(k,n) = 0  for all n  (and this is a limit of an eventuallyconstant sequence).  Surely set theory must be to blame for this!> 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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