Re: abundance of irrationals!)

In article <fb701d3c.0504091328.686ed77a@xxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx (W. Mueckenheim) writes:
> "*** T. Winter" <***.Winter@xxxxxx> wrote in message news:<IELt4E.K2G@xxxxxx>...
....
> > The above is not a series. You have first to *define* what the sum over
> > k in N is meaning. So long as you do not define it it is nothing. So
> > what do you *mean* with the sum over k in N? The standard meaning is:
> > lim{n -> oo} sum {k <= n} 1/2^k.
>
> The standard meaning of Cauchy's convergence criterion is to find out
> something for ALL n > n_0. This implies an infinite sum, but not the
> limit n-->oo the existence of which you want to establish by using
> Cauchy's criterion.

How do you come at that stupid idea? You have apparently no notion about
the definition of the concept of limits.

> > > It is incredible! Cauchy's convergence criterion is wrong! Mathematics
> > > has been wrong for 200 years!
> >
> > In what way is Cauchy's convergence criterion wrong?
>
> You oppose to infinite sums but you need a proof for ALL n > n_0 which
> ultimately leads to an infinite sum, before you know whether the
> series is converging.

How wrong you are. Infinite sums do *not* exist (there is no theorem
or axiom that gives meaning to them). In the case:
sum{k=1...n} 1/2^k = 1 - 1/2^k
so by the *definition* of limit the series converges and its limit is 1.
Also you do not understand Cauchy's convergence criterion. It does not
talk about sums, but only about differences between pairs of elements.

You have still some elementary mathematics to learn.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.