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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 17, 2009 6:45 AM

In article <GLidnXhuo5t347TWnZ2dnUVZ_sWdnZ2d@giganews.com> "K_h" <KHolmes@SX729.com> writes:
> "Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
> news:KuqwG3.qv@cwi.nl...

...
> > > > The definition you provided for a sequence of sets A_n
> > > > depends on whether
> > > > each A_n is or is not a set containing a single set as
> > > > an
> > > > element.
> > > >
> > > > Your definition leads to some strange consequences. I
> > > > can
> > > > state the
> > > > following theorem:
> > > >
> > > > Let A_n and B_n be two sequences of sets. Let A_s =
> > > > lim
> > > > sup A_n and
> > > > A_i = lim inf A_n, similar for B_s and B_i. Let C_n
> > > > be
> > > > the sequence
> > > > defined as:
> > > > C_2n = A_n
> > > > C_(2n+1) = B_n
> > > > Theorem:
> > > > lim sup C_n = union (A_s, B_s)
> > > > lim inf C_n = intersect (A_i, B_i)
> > > > Proof:
> > > > easy.

...
> > Well, the above theorem is still not valid with your
> > definition.

>
> What case did you have in mind? I found cases where it
> works fine.

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}.
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).
--
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