```Date: Dec 18, 2009 9:26 AM
Author: Dik T. Winter
Subject: Re: Another AC anomaly?

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}, butlim B_n equals {N}, where B_n is a subsequence of C_n.  Strangethat an infinite subsequence can have a limit different from thelimit of the original sequence.-- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/
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