On 14 Okt., 22:11, SPQR <S...@roman.gov> wrote: > In article > <fe4017a8-ad88-4946-8f93-8e4f32bc2...@a12g2000vbz.googlegroups.com>, > > > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > On 14 Okt., 09:27, Virgil <vir...@ligriv.com> wrote: > > > > > > It is quite possible to define ant two of H(n), B(n) and R(n) so that > > > > > they are equal for all finite naturals n, but necessarily have different > > > > > limit values as n -> oo. > > > > > Of course. That is possible for every sequence or series. But that > > > > would kill mathematics of limits. > > > > Not at all. There is nothing in standard mathematics that requires every > > > infinite sequence to have a limit. > > > I agree. In particular the so called limit ordinal has nothing at all > > to do with mathematics. Therefore there is no limit of the sequence > > (n). > > > > If Wm wants his mathematics to be different, he cannot expect everyone > > > to kowtow to his demands. > > > But it should be expectet that mathemticians can understand a bit of > > logic. For example, when I say that three identical sequences, if they > > have a limit, must have the same limit. > > Except that the three sequences are not identical, since two of them > have a common fixed point and the third has no fixed points.
Two sequences that are identical for every finite n simply are the same sequences. Here we have three sequences B(n) = H(n) = R(n). If one of them has a limit, then all have limits. If one has a maximum, then all have maxima. But we know that at least one of them has not a maximum. Therefore none has a maximum. And if one has no limit at all, then all have not limits.
That all has not even to do with triangles and edges and "fixed points", but is simply a consequence of mathematics of infinite sequences.