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Topic: Another AC anomaly?
Replies: 43   Last Post: Dec 21, 2009 8:08 AM

 Messages: [ Previous | Next ]
 Dik T. Winter Posts: 7,899 Registered: 12/6/04
Re: Another AC anomaly?
Posted: Dec 18, 2009 9:26 AM

In article <jrydnSyLVZ_6vbbWnZ2dnUVZ_vOdnZ2d@giganews.com> "K_h" <KHolmes@SX729.com> writes:
> "Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
> news:Kusoo8.xz@cwi.nl...

...
> > Let's have some arbitrary object 'a' and the natural
> > numbers. Create
> > the sequence A_n where A_n = {a} and the sequence B_n
> > where B_n = {n}.
> > According to your definition:
> > lim sup A_n = {a}
> > and
> > lim sup B_n = {N}.
> > Now create the sequence C_n: C_2n = A_n, C_2n+1 = B_n.
> > Again according
> > lim sup C_n = {a}
> > which is not equal to union (lim sup A_n, lim sup B_n).

>
> This is a good example, thanks. Your theorem only applies
> in special cases for the definition I have offered (although
> my definition satisfies some different but interesting
> theorems).

Such as? Certainly not:
limsup | S_n | = |limsup S_n|
because see for that the sequence C_n above and limsup. Stranger,
with your definition, lim C_n does exist and is equal to {a}, but
lim B_n equals {N}, where B_n is a subsequence of C_n. Strange
that an infinite subsequence can have a limit different from the
limit of the original sequence.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Date Subject Author
12/12/09 Jesse F. Hughes
12/13/09 K_h
12/14/09 Dik T. Winter
12/14/09 K_h
12/15/09 Dik T. Winter
12/15/09 K_h
12/16/09 Dik T. Winter
12/16/09 K_h
12/17/09 Dik T. Winter
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/21/09 Dik T. Winter
12/14/09 Dik T. Winter
12/14/09 K_h
12/15/09 Dik T. Winter
12/15/09 Dik T. Winter
12/15/09 K_h
12/16/09 Dik T. Winter
12/17/09 K_h
12/17/09 Dik T. Winter
12/18/09 K_h
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/15/09 K_h
12/16/09 Jesse F. Hughes
12/17/09 Dik T. Winter
12/17/09 Jesse F. Hughes
12/16/09 Dik T. Winter
12/15/09 ross.finlayson@gmail.com
12/13/09 K_h
12/13/09 Jesse F. Hughes
12/13/09 Jesse F. Hughes
12/14/09 Ilmari Karonen
12/14/09 Jesse F. Hughes
12/15/09 Chas Brown