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Topic: Another AC anomaly?
Replies: 280   Last Post: Dec 27, 2009 3:53 PM

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K_h

Posts: 419
Registered: 4/12/07
Re: Another AC anomaly?
Posted: Dec 11, 2009 9:39 PM
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"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
news:KuHL7p.3zE@cwi.nl...
> In article <5ZedndTmvoBac7zWnZ2dnUVZ_qOdnZ2d@giganews.com>
> "K_h" <KHolmes@SX729.com> writes:

> > "Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
> > news:KuFy3L.Cxt@cwi.nl...

> ...
> > > Have a look at <http://en.wikipedia.org/wiki/Lim_inf>
> > > in
> > > the section
> > > titled "Special case: dicrete metric". An example is
> > > given with the
> > > sequence {0}, {1}, {0}, {1}, ...
> > > where lim sup is {0, 1} and lim inf is {}.
> > >
> > > Moreover, in what way can a definition be invalid?

> >
> > It depends on the context. When it comes to supertasks,
> > limsup={0,1} is basically useless. That is why those
> > definitions are not good, and invalid, for evaluating
> > supertasks -- in response to WM's supertask issues.

>
> I am not discussing supertasks, nor is WM here. The
> question is simply
> whether it is possible that:
> lim | S_n | != | lim S_n |
> with some form of limit. And by the definitions I gave
> (and which you
> also will find on the wikipedia page above):
> lim(n -> oo) {n} = {}


The only way lim(n ->oo){n}={} is if limsup and liminf both
equal {}. If limsup and liminf are different then the limit
does not exist and cannot equal an existing set like {}.
Since the empty set exists, and since you are claiming that
lim(n ->oo){n}={}, you need to show that limsup and liminf
are both {} by the definition you are using. WM attributed
this definition to you:

- Given a sequence of sets S_n then:
- lim sup{n -> oo} S_n contains those elements that occur in
infinitely many S_n.
- lim inf{n -> oo} S_n contains those elements that occur in
all S_n from a certain S_n (which can be different for each
element).
- lim{n -> oo} S_n exists whenever lim sup and lim inf are
equal.

All the definitions on the wikipedia pages, and the above
definition WM attributes to you, give lim(n ->oo){n}=N.
Consider the naturals:

S_0 = 0 = {}
S_1 = 1 = {0}
S_2 = 2 = {0,1}
S_3 = 3 = {0,1,2}
S_4 = 4 = {0,1,2,3}
S_5 = 5 = {0,1,2,3,4}
...

First, let's consider the definition WM attributes to you:

- lim sup{n -> oo} S_n contains those elements that occur in
infinitely many S_n.

* Every natural, n, occurs infinitely many times after S_n
so limsup=N.

- lim inf{n -> oo} S_n contains those elements that occur in
all S_n from a certain S_n (which can be different for each
element).

* Every natural, n, occurs in each S_m after S_n so
liminf=N.

- lim{n -> oo} S_n exists whenever lim sup and lim inf are
equal.

* limsup=liminf=N and so the limit exists and is N.

On the wikipedia page, apply the definition under the
section titled "General set convergence" and apply both
definitions under the section titled "special case: discrete
metric" to the above S_n and you will find that the limit is
N in all cases. So, in all cases we get lim|S_n|=|limS_n|.
My previous post already has the proof for wikipedia's last
definition:

liminf S_n=\/(n=0, oo)[/\(m=n, oo) S_m]
(n-->oo)
limsup S_n=/\(n=0, oo)[\/(m=n, oo) S_m]
(n-->oo)

In fact, the wikipedia page says the definitions are
equivalent in certain cases: "The difference between the
two definitions involves the topology (i.e., how to quantify
separation) is defined. In fact, the second definition is
identical to the first when the discrete metric is used to
induce the topology on X."

k





Date Subject Author
11/23/09
Read Re: Another AC anomaly?
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11/23/09
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11/30/09
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12/1/09
Read Jesse Hughes of Bentley College uses signature block to defame
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plutonium.archimedes@gmail.com
12/1/09
Read Re: Jesse Hughes of Bentley College uses signature block to defame
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12/1/09
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12/1/09
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of Bentley College
Marshall
12/2/09
Read Re: using signature-blocks to defame innocent people Re: Jesse Hughes
of Bentley College
Marshall
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
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William Hughes
11/26/09
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mueckenh@rz.fh-augsburg.de
11/28/09
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mueckenh@rz.fh-augsburg.de
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11/26/09
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11/28/09
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11/26/09
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mueckenh@rz.fh-augsburg.de
11/28/09
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11/28/09
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mueckenh@rz.fh-augsburg.de
11/29/09
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Virgil
11/29/09
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ross.finlayson@gmail.com
11/29/09
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Marshall
11/30/09
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Virgil
11/30/09
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ross.finlayson@gmail.com
11/30/09
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ross.finlayson@gmail.com
11/30/09
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11/30/09
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ross.finlayson@gmail.com
11/30/09
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Virgil
11/30/09
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ross.finlayson@gmail.com
11/30/09
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11/29/09
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mueckenh@rz.fh-augsburg.de
11/26/09
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11/26/09
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11/26/09
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11/29/09
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11/27/09
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11/27/09
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11/30/09
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11/30/09
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11/27/09
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11/27/09
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mueckenh@rz.fh-augsburg.de
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11/27/09
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11/27/09
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11/27/09
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11/27/09
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11/27/09
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11/27/09
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mueckenh@rz.fh-augsburg.de
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11/30/09
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mueckenh@rz.fh-augsburg.de
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mueckenh@rz.fh-augsburg.de
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mueckenh@rz.fh-augsburg.de
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mueckenh@rz.fh-augsburg.de
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12/7/09
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12/18/09
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12/18/09
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12/19/09
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12/19/09
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12/22/09
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12/22/09
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12/18/09
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12/21/09
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12/22/09
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12/27/09
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12/16/09
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12/11/09
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12/11/09
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12/11/09
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12/11/09
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12/12/09
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12/10/09
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12/8/09
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12/16/09
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12/16/09
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12/4/09
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12/7/09
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12/8/09
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11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil

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