K_h
Posts:
419
Registered:
4/12/07


Re: Another AC anomaly?
Posted:
Dec 18, 2009 1:25 AM


"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message news:Kusoo8.xz@cwi.nl... > 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}. > 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).
k

