On Nov 22, 12:06 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > 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?
It answers it.
<I explicitly used > alternating sequences 010101...
And I deal with the simpler case first and then with the more complicated alternating digit case. Note
This 1 corresponds to the digit with index 5. We conclude that for each index the digit corresponding to the [index] does not exist in the limit.
This covers the alternating digit case and a lot of other cases.