Re: Question

In article <1146163021.981085.118390@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
zuhair <zaljohar@xxxxxxxxx> wrote:

Arturo Magidin wrote:

Look at the DEFINITION:

A sequence of rationals is function from the natural numbers to the
rationals.

We say a sequence (a_0, a_1,...) (usually denoted {a_i}) converges to
the rational number Q if and only for every N>0 there exists M>0 such
that if n>M, then |a_n-Q|<1/N.

still there is a problem, when you say converges to zero,

No, there is no problem when I say that. I am going EXACTLY by the
definition.

You may, perhaps, object to the definition, but then you are objecting
to all of mathematics.

at what
number of terms in that series this will happen,

This question is nonsense. I did NOT say that the sequence is
"eventually equal to zero". That is something else. I said the
sequence CONVERGES to zero. That is something else, and was EXPLICITLY
and CLEARLY defined for you above.

is it at Omega of c_i
or Omega+1 or 2^Omega.

None of them. There is no omega here. Sequences are indexed by NATURAL
numbers. Natural numbers are, BY DEFINITION, finite. Omega is NOT a
natural number. "The sequence converges to zero" is NOT equivalent to
"the sequence is eventually equal to zero".


Your Cauchy priniciple didn't mention that.

Yes, it did. It gave ALL the definitions. You just seem unable to
parse them.

A sequence of rationals was defined to be a function from the NATURALS
to the rationals. We write it as (a_0, a_1, a_2, ..., a_n, ...), where
"a_n" is the value of the function at the natural number n. The only
terms that are defined are the terms indexed by NATURAL numbers, which
are FINITE. There is no "omega", there is no "omega + 1", there is no
"2^omega". There are only finite numbers.

The definition of "cauchy sequence" and "convergence" relates ONLY to
sequences; thus, it relates ONLY to finite indices. The definition of
convergence and of cauchy sequence is given entirely in terms of
FINITE indices and conditions on them.

Period.

It doesn't differentiate
between
0.9999........
which contains Omega of nines in it after the decimal point.


and 0.9999....... 9 which contains Omega+1 of nines in it after the
decimal point.

That expression does NOT represent a sequence, so it is not under
consideration. It is merely a symbolic string of symbols which does
not represent a real number.

In reality I tend to think that it is at what Cantor called once as
"The Absolute Infinity" number of terms in those series that c_i will
be zero

In reality I tend to think that you are not bothering to
think. Certainly, you seem either unwilling or incapable of reading
simple mathematical definitions.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.

Relevant Pages

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