Re: abundance of irrationals!)

Paul Murray <paul@xxxxxxxxxx> wrote in message news:<43y7e.8197315$f47.1498594@xxxxxxxxxxxxxxxxx>...
> In article <fb701d3c.0504140832.7716f82b@xxxxxxxxxxxxxxxxxx>, W. Mueckenheim wrote:
> > Correct. But could you explain the difference between "all n are
> > included in at least one of the sums" and "at least one of the sums
> > includes ALL n". I can't see any.
>
> 1) For every possible value of n, you can find at least one sum which
> includes it
>
> 2) You can find at least one sum which includes every possible value of n
>
> Those are very different statements.
>
> Simple example:
>
> S = { 1, 2, 3 }
>
> It is *true* that: For all s1 in S, there exists s2 in S such that s1 != s2
> It is *false* that: There exists s2 in S such that for all s1 in S, s1 != s2
>
> Exactly the same words in a different order, and they meant different things.


Yes, but your example is missing the point, i.e. the SUM in a linearly ordered set.
Form the (converging) SUM a_k over k = 1 to n.
Do this for every finite n, without leaving out any.
This implies that one of the sums includes all n e N.

If you disagree, tell me which n is left out.

Regards, WM
.

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