Am Sonntag, 8. Dezember 2013 02:28:10 UTC+1 schrieb Virgil:
> Note that for every n in |N there will be infinitely many rationals > > whose first n digits agree with d and only differ later.
Of course. This is a fact since every decimal representation of a number is nothing but the set of FIS. So up to every d_n there is a FIS. And a FIS is a rational number. > > > > So that there can never be any n such that. by digit n. d will differ > > from all rationals.
Simply because d has no complete decimal representation. > > > > Nevertheless any irrational, of which even WM admits there are > > infinitely many, will differ from every rational in infinitely many > > places.
No. It will differ from every "given" rational. But most rationals cannot be given, because only the tiny set of the first rationals is available to supply "given" rationals. Most rationals belong to the inaccessible set beyond any "given" one.