K_h
Posts:
419
Registered:
4/12/07


Re: Another AC anomaly?
Posted:
Dec 14, 2009 8:05 PM


"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message news:KunF0v.9y9@cwi.nl... > In article <k9mdnWn11pFIAbjWnZ2dnUVZ_uWdnZ2d@giganews.com> > "K_h" <KHolmes@SX729.com> writes: > > > > let L=lim(n>oo)X_n be the specified wikipedia limit > > for > > X_n. If L exists then: > > > > lim(n>oo)A_n = lim(n>oo){X_n} = {L} > > > > otherwise lim(n>oo)A_n = lim(n>oo){X_n} does not > > exist. > > Under this definition lim(n>oo){n}={N} > > By what definition is it {N}?
If a sequence of sets, A_n, cannot be expressed as {X_n}, for some sequence of sets X_n, then lim(n>oo)A_n is defined by one of the two wikipedia limits. Otherwise let L=Wikilim(n>oo)X_n be the specified wikipedia limit for X_n. If L exists then define lim(n>oo)A_n=lim(n>oo){X_n} as follows:
lim(n>oo)A_n = lim(n>oo){X_n} = {L}
otherwise lim(n>oo)A_n = lim(n>oo){X_n} does not exist. I referenced the wikipedia limit here just to save time. Which wikipedia definition is selected is up to the user. If you feel that defining a limit in terms of another limit definition is bad taste then the above definition could be easily reworded to include the relevant material without reference to wikipedia.
> By what definition is: > lim(n > oo) n = N?
Any of the wikipedia definitions. Here is the proof again.
Use this definition:  Given a sequence of sets S_n then:  lim sup{n > oo} S_n contains those elements that occur in infinitely many S_n.  lim inf{n > oo} S_n contains those elements that occur in all S_n from a certain S_n (which can be different for each element).  lim{n > oo} S_n exists whenever lim sup and lim inf are equal.
Theorem: lim(n >oo) n = N. Consider the naturals:
S_0 = 0 = {} S_1 = 1 = {0} S_2 = 2 = {0,1} S_3 = 3 = {0,1,2} S_4 = 4 = {0,1,2,3} S_5 = 5 = {0,1,2,3,4} ... S_n = n = {0,1,2,3,4,5,...,n1} ... S_N = N = {0,1,2,3,4,5,...,n1,n,n+1...}
 lim sup{n > oo} S_n contains those elements that occur in infinitely many S_n.
* Every natural, n, occurs infinitely many times after S_n so limsup=N.
 lim inf{n > oo} S_n contains those elements that occur in all S_n from a certain S_n (which can be different for each element).
* Every natural, n, occurs infinitely many times after S_n so liminf=N.
 lim{n > oo} S_n exists whenever lim sup and lim inf are equal.
* limsup=liminf=N and so the lim(n >oo)n exists and is N.
k

