LudovicoVan
Posts:
2,971
From:
London
Registered:
2/8/08
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Re: Matheology § 152
Posted:
Nov 16, 2012 2:42 PM
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"WM" <mueckenh@rz.fh-augsburg.de> wrote in message
news:f551d5a6-d25a-4509-8d16-77200b1038a5@l12g2000vbj.googlegroups.com...
> "LudovicoVan" <ju...@diegidio.name> wrote:
>
> >> What is the limit of the sequence of the sets of indexes on the left
> >> hand side?
>
> > {oo}
>
> oo is not a natural number but indices are natural numbers because
> they determine a position. oo is not a position.
Was that an objection? We are just using limits as usual.
Let's make it formal, with the original balls and vase problem where at each
step we simply put 10 new balls in and remove the oldest one.
That translates to a sequence of sets, in fact a sequence of intervals of
natural numbers so defined:
s(n) := { i in N | n+1 <= i <= 10*n } =
= [ n+1; 10*n ]
For example, that gives:
s(0) := { } // empty
s(1) := [ 2; 10 ]
s(2) := [ 3; 20 ]
s(2) := [ 4; 30 ]
etc.
The limit of that sequence is:
lim_{n->oo} s(n) =
= lim_{n->oo} [ n+1; 10*n ] =
= [ lim_{n->oo} n+1; lim_{n->oo} 10*n ] =
= [ oo; oo ] =
= { oo }
> >> What is the limit of the decimal numbers?
>
> > oo
>
> > By the same basic arithmetic: note that if you disallow the first,
> > you should disallow the second on exactly the same grounds.
>
> Talking about the limit oo does not mean that it is assumed. It means
> that the sequence grows beyond every finite positive number.
>
> Also the sequence (1/n) does never assume its limit 0. Only the
> difference between 1/n and 0 shrinks below every positive number.
Just the same as above.
-LV
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