K_h
Posts:
419
Registered:
4/12/07


Re: Another AC anomaly?
Posted:
Dec 15, 2009 10:00 PM


"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message news:Kup2G0.IBK@cwi.nl... > In article <iNWdnfmPh7NpQ7vWnZ2dnUVZ_hydnZ2d@giganews.com> > "K_h" <KHolmes@SX729.com> writes: > > "Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message > > news:KunEFz.913@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.
Yes, my definition did not include a limsup and liminf but they can be added. With this addition, the limit of sets like {X_n} is more in line with the general notion of a limit.
If a sequence of sets, A_n, cannot be expressed as {X_n}, for some sequence of sets X_n, then limsup{n>oo}A_n, liminf{n>oo}A_n, and lim(n>oo)A_n are defined by a specified wikipedia limit. Otherwise, let X_s, X_i, and L be the specified supremum, infimum, and limit (defined by wikipedia) on X_n. Define
limsup{n>oo}A_n = {X_s} liminf{n>oo}A_n = {X_i}
If L exists then lim(n>oo)A_n = {L} otherwise it does not exist.
k

