The Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
K_h

Posts: 419
Registered: 4/12/07
Re: Another AC anomaly?
Posted: Dec 14, 2009 8:05 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply


"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
Read Re: Another AC anomaly?
Jesse F. Hughes
12/13/09
Read Re: Another AC anomaly?
K_h
12/14/09
Read Re: Another AC anomaly?
Dik T. Winter
12/14/09
Read Re: Another AC anomaly?
K_h
12/15/09
Read Re: Another AC anomaly?
Dik T. Winter
12/15/09
Read Re: Another AC anomaly?
K_h
12/16/09
Read Re: Another AC anomaly?
Dik T. Winter
12/16/09
Read Re: Another AC anomaly?
K_h
12/17/09
Read Re: Another AC anomaly?
Dik T. Winter
12/18/09
Read Re: Another AC anomaly?
K_h
12/18/09
Read Re: Another AC anomaly?
Dik T. Winter
12/18/09
Read Re: Another AC anomaly?
K_h
12/21/09
Read Re: Another AC anomaly?
Dik T. Winter
12/14/09
Read Re: Another AC anomaly?
Dik T. Winter
12/14/09
Read Re: Another AC anomaly?
K_h
12/15/09
Read Re: Another AC anomaly?
Dik T. Winter
12/15/09
Read Re: Another AC anomaly?
Dik T. Winter
12/15/09
Read Re: Another AC anomaly?
K_h
12/16/09
Read Re: Another AC anomaly?
Dik T. Winter
12/17/09
Read Re: Another AC anomaly?
K_h
12/17/09
Read Re: Another AC anomaly?
Dik T. Winter
12/18/09
Read Re: Another AC anomaly?
K_h
12/18/09
Read Re: Another AC anomaly?
K_h
12/18/09
Read Re: Another AC anomaly?
Dik T. Winter
12/18/09
Read Re: Another AC anomaly?
K_h
12/18/09
Read Re: Another AC anomaly?
Dik T. Winter
12/18/09
Read Re: Another AC anomaly?
K_h
12/15/09
Read Re: Another AC anomaly?
K_h
12/16/09
Read Re: Another AC anomaly?
Jesse F. Hughes
12/17/09
Read Re: Another AC anomaly?
Dik T. Winter
12/17/09
Read Re: Another AC anomaly?
Jesse F. Hughes
12/16/09
Read Re: Another AC anomaly?
Dik T. Winter
12/15/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
12/13/09
Read Re: Another AC anomaly?
K_h
12/13/09
Read Re: Another AC anomaly?
Jesse F. Hughes
12/13/09
Read Re: Another AC anomaly?
Jesse F. Hughes
12/14/09
Read Re: Another AC anomaly?
Ilmari Karonen
12/14/09
Read Re: Another AC anomaly?
Jesse F. Hughes
12/15/09
Read Re: Another AC anomaly?
Chas Brown

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2017. All Rights Reserved.