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Topic: Another AC anomaly?
Replies: 43   Last Post: Dec 21, 2009 8:08 AM

 Messages: [ Previous | Next ]
 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,...,n-1}
...
S_N = N = {0,1,2,3,4,5,...,n-1,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

Date Subject Author
12/12/09 Jesse F. Hughes
12/13/09 K_h
12/14/09 Dik T. Winter
12/14/09 K_h
12/15/09 Dik T. Winter
12/15/09 K_h
12/16/09 Dik T. Winter
12/16/09 K_h
12/17/09 Dik T. Winter
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/21/09 Dik T. Winter
12/14/09 Dik T. Winter
12/14/09 K_h
12/15/09 Dik T. Winter
12/15/09 Dik T. Winter
12/15/09 K_h
12/16/09 Dik T. Winter
12/17/09 K_h
12/17/09 Dik T. Winter
12/18/09 K_h
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/15/09 K_h
12/16/09 Jesse F. Hughes
12/17/09 Dik T. Winter
12/17/09 Jesse F. Hughes
12/16/09 Dik T. Winter
12/15/09 ross.finlayson@gmail.com
12/13/09 K_h
12/13/09 Jesse F. Hughes
12/13/09 Jesse F. Hughes
12/14/09 Ilmari Karonen
12/14/09 Jesse F. Hughes
12/15/09 Chas Brown