
Re: Another AC anomaly?
Posted:
Dec 18, 2009 9:26 AM


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.  dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

