On 22 Nov., 16:27, William Hughes <wpihug...@gmail.com> wrote: > On Nov 22, 3:30 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > Can we estimate by means of set theory how many digits *left* to the > > decimal point will be present in the limit (as calculated by > > analytical means) of the real sequence > > > > > 01. > > > > 0.1 > > > > 010.1 > > > > 01.01 > > > > 0101.01 > > > > 010.101 > > > > 01010.101 > > > > 0101.0101 > > > > ... > > > ?
> Yes, The set of digits left of the decimal point is the > empty set.
This is in contradiction to analysis (although analysis is said to be based upon set theory). Just my point.
> Simplest argument. Start with > > 100.000... > 10.000... > 1.000... > 0.1000... > 0.01000... > ... > > The 1 does not exist in the limit. This 1 corresponds to > the digit with index 5. We conclude that for > each index the digit corresponding to the digit does > not exist in the limit. Thus the set of digits in the limit > is the empty set. Thus, in the limit, the set of digits to > the left of the decimal point is the empty set.
What has this problem to do with my question? I explicitly used alternating sequences 010101... (moving from left to right, - so your next example is completetly off topic). Analysis gives a result. Set theory gives another result which is incompatible with analysis.