Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 K_h Posts: 419 Registered: 4/12/07
Re: Another AC anomaly?
Posted: Dec 15, 2009 10:00 PM

"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.
>
> 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
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 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