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

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K_h

Posts: 419
Registered: 4/12/07
Re: Another AC anomaly?
Posted: Dec 15, 2009 10:00 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply


"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
news:Kup2G0.IBK@cwi.nl...
> In article <iNWdnfmPh7NpQ7vWnZ2dnUVZ_hydnZ2d@giganews.com>
> "K_h" <KHolmes@SX729.com> writes:

> > "Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
> > news:KunEFz.913@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.


Yes, my definition did not include a limsup and liminf but
they can be added. With this addition, the limit of sets
like {X_n} is more in line with the general notion of a
limit.

If a sequence of sets, A_n, cannot be expressed as {X_n},
for some sequence of sets X_n, then limsup{n-->oo}A_n,
liminf{n-->oo}A_n, and lim(n-->oo)A_n are defined by a
specified wikipedia limit. Otherwise, let X_s, X_i, and L
be the specified supremum, infimum, and limit (defined by
wikipedia) on X_n. Define

limsup{n-->oo}A_n = {X_s}
liminf{n-->oo}A_n = {X_i}

If L exists then lim(n-->oo)A_n = {L} otherwise it does not
exist.

k




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

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