This sequence grows without limit. > > > This can be proved by taking any number n and showing > > that there is a number k such that all for terms a(j) of the sequence > > with k > j we have a(j) > n. Proof: For given n take k = n + 10. > > ow does that work for the sequence a(j) = 0 for all j?
Is 0 larger than any number n?
> > > Every set theorist knows that the sequence of sets of indices left of > > the decimal point has the limit empty set. This is an requirement of > > set theory. > > Then let us see which axiom, or set of axioms, of some set theory which > actually requires such nonsense. say among the axioms for ZFC, for > example.
Try to learn it. Look what William Hughes just explains here. > > > > > And finally everybody knows that decimal numbers, by definition, > > cannot consist of digits that have no indexs. > > Numbers (decimal or otherwise) can exist without any digits of any sort, > but decimal numerals can not.
But the numbers in above list exist with their digits. > > Since a numeral is merely a name for a number,