Re: Question

In article <1146674942.041712.253680@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"zuhair" <zaljohar@xxxxxxxxx> wrote:

Virgil wrote:
In article <MPG.1ec18fde4ae012a698acaa@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil said:
In article <1146562841.910319.71390@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"zuhair" <zaljohar@xxxxxxxxx> wrote:


Every real can be represented by a convergent sequence of rationals
in
the Cauchy model and the representation can be chosen in a variety
of
ways, including the one in which 0.999... is regarded as the limit
(which can be proved to exist) of the sequence, 0.9, 0.99, 0.999,
....

Zuhair should go join T. Orlow to make perfect a folie a duex.

I agree! and disagree .

0.9999..... is not the limit of the sequence 0.9 ,0.99,0.999,.... ,
it
is the upper bound of that sequence.

There is no such thing as "the" upper bound. It is "the" least upper
bound, and since the sequence in increasing and bounded, the LUB equals
the limit, which must exist in standard mathematics.

One wonders what sort of non-standard mathematics Zuhair thinks he can
make work in which a bounded increasing sequence does NOT converge to
its LUB>

Number 1.000..... is the limit of that sequnce.

But by then you should define the world limit in a clear manner.


What in the world is the "world limit"?

I think he meant, 'you should define the word "limit" in a clear manner'.
Either that or it's something about the surface of the Earth.


A sequence is, by definition, a function whose domain is the set of
natural numbers N.

A real sequence is, by definition, a sequence whose codomain is the set
of reals, R.

A real sequence s:N -> R, has limit L in in R if and only if for every
positive epsilon in R, the set {n: |s(n) - L| >= epsilon} is finite.

So, does one have to use the word, "hypersequence", to denote one that is
uncountably long, with infinite indexes on elements? I still can't
understand
how this can be proscribed.

Once "sequence" has been defined in mathematics, and it has been, one
must not use the word "sequence" in mathematics to refer to anything
which does not fit that definition.

One must either invent a new word, or find one with no mathematically
defined meaning, to use for any other purpose.


If what you mean by lim ( 0.9 ,0.99,0.999,.... ) is the smallest REAL
number that is bigger than all terms belonging to the sequnce 0.9
,0.99,0.999,.... . , then: lim ( 0.9 ,0.99,0.999,.... ) =
1.000......... no doubt.

That is not precisely what I mean by a limit, see above, but it is true
that the LUB, if one exists, for an increasing real sequence is also
its
limit.


Is the limit of the naturals omega?

No, but the union of them is, provided one uses the von Neumann naturals.

Wrong.

Since each vN natural is a subset and member of the next, the union of
all of them contains all of them, and that is precisely omega.

Zuhair has this hair up his xxx that he can remake mathematics without
let or hindrance. He and TO make a pair of abject failures.
.

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