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, 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

>