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}, 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/