In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 14 Okt., 13:32, William Hughes <wpihug...@gmail.com> wrote: > > > > If there is a complete infinite set |N, then it has not a last > > > element. > > > Then only two of three identical (for every n) sequences have limits > > > that are maxima whereas the third limit is only a supremum without > > > being a maximum. > > > > Correct, in other words two of the limits are part of the > > "infinite" triangle and one is not. > > Yes, *then*. > But in mathematics, identical sequences have identical limits. > Therefore either all limits are maxima (false) or all limits are > suprema but not maxima (true). > > Of course aleph_0 as the maximum R(oo) would imply that also the lines > have a maximum. > > This would in particular follow from the construction equivalent to a > certain rotating method of counting all rational numbers, namely > > 1 > > 1 > 22 > > 3 > 31 > 322 > > 4 > 34 > 314 > 3224 > > 4 > 34 > 314 > 3224 > 55555 > > and so on, where nnn...n denotes the edge added in the n-th step. > There is no discriminated edge possible. > > Therefore the diagonal edge is not larger than the other edges.
Thus WM has invented a miraculously equilateral right triangle. --