In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 10 Dez., 21:16, Virgil <vir...@ligriv.com> wrote: > > In article > > <f659932b-e97c-4051-897a-8f0ea6688...@f17g2000vbz.googlegroups.com>,
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > >> Further in ZFC the sequence > > > > >> 21., 2.1, 432.1, 43.21, 6543.21, 654.321, ... > > > > >> has the limit < 1. > > > > Does your sequence continue on with digits larger than 9 or does digit 0 > > follow digit 9 cyclicly, or does it do something entirely different at > > that point? > > That does not matter for my result. > But obviously we go on with multi-digit natural numbers.
Nothing with non-zero digits to the right oF the radix point are natural numbers, at least not outside Wolkenmuekenheim. > > You should know that a simpler sequence replacing even and odd numbers > by 0 and 1, repsectively, shows the same result.
Anything that WM alone claims needs more proof that just his word to convince anyone sensible. But what WM presents as proof almost always fail to convince the careful reader. > > > > As it stands, your sequence is far too ambiguous to have any defineable > > or determinable limit. > > No. Every mathematician can determine the limit - not you, of course.
Until one has an unambiguous rule determining the terms of a sequence, which does not exist in WM's "sequence", there cannot be an unambiguous limit. > > > > > > > > > > No; it has no limit. > > > > > It has an improper limit in analysis, namely oo. It has a limit in set > > > theory.
Until WM can show that his sequence of numbers is well defined, it has no limit as a sequence of numbers, and until WM can show that his sequence of numbers is really a sequence of sets, which he has not done, it certainly cannot have a set limit.
And, quite probably, even if WM finally does show both of those things, it may be that neither limit will be defined by any standard limit process, either as a number sequence or a set sequence. > > > > Unless it is redefined as a sequence of sets, which it is not in its > > original form, set theory says nothing about any limit for it. > > Sets of numbers are sets, whether or not they appear in my sequence of > real numbers. There is no redifinition necessary.
The only set of numbers involved would be the set of all numbers in the sequence and that is not a sequence of sets. > > > > Please do not mistake analysis and set theory. In analysis we have the > > > simple criterion: If lim 1/a_n = 0, then lim a_n = oo. That is here > > > the case. Obviously. > > > > Not until you explain what happens to the terms of your as yet ambiguous > > sequence when you need to use digits following 9. > > Until ,then nothing about it is obvious. > > Only if blind men try to read it. But I assume that Seymour J. Shmuel > is not blind. So wait what he will have to say.
To the best of my ability to see, there is no pattern of terms so obvious as to rule out all other possibilities once the present pattern extends far enough to needs a successor to digit 9.
Perhaps WM will have pity on my weak vision and explain his rule for that sequnce less ambiguously. --