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 <> 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, 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!

> 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