Search All of the Math Forum:

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

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 18, 2009 9:07 PM

"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
news:Kuuqrs.FJ7@cwi.nl...
> 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
> > > to your definition:
> > > 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.

Let A_n and B_n be two sequences of sets of the form {X_n}.
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 and
C_(2n+1) = B_n.

Theorem:
Since A_s = {a_s} and B_s = {b_s}

lim sup C_n = {a_s \/ b_s}

lim inf C_n = {a_i /\ b_i}

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