On 23 Okt., 17:55, FredJeffries <fredjeffr...@gmail.com> wrote: > On Oct 23, 1:27 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > There is no reason that these limits exist. > > But if they exist, then there is no chance to give differing answers. > > At least, there is no reason with foundation in mathematics. In > > mathemtics limits are completely defined by the finite terms. > > The limit of a sequence is determined by the tail(s) of the > sequence.
Every element of the sequence including every element of every tail has a finite index. That is, every element is a finite term.
But all that is of little interest for the following argument:
If you accept that the sequence
1 1,2 1,2,3 ...
has limit omega, then you cannot accept, that it sometimes has the limit as a maximum and sometimes has the limit not as maximum, but only as supremum.
For that sake, we need no topology. From Archimed to Cantor there were limits calculated without knowing what a topology is. But if you cannot compute limits without topology, then take just that topology that gives you the limit omega for the sequence of sets
1 1,2 1,2,3 ...
and try to find, in the same topolgy, the limit of the sequence of sets
1 2,3 4,5,6 ...
You should get the empty set.
And finally use the first enumeration for superscripts and the second for subscripts. And then find in the same topology the limit. Good luck!
