Re: abundance of irrationals!)

"*** T. Winter" <***.Winter@xxxxxx> wrote in message news:<IEr61u.GHJ@xxxxxx>...
> In article <IEr5A8.Evx@xxxxxx> "*** T. Winter" <***.Winter@xxxxxx> writes:
> > In article <fb701d3c.0504091328.686ed77a@xxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx (W. Mueckenheim) writes:
> ...
> > > 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.
>
> Sorry, this is wrong (of course). It talks about differences of finite
> sums.

It talks about "ALL n" > n_0. Do you think "ALL n" supply a finite sum?

Regards, WM
.