On Nov 21, 1:57 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 21 Nov., 18:43, William Hughes <wpihug...@gmail.com> wrote: > > > On Nov 21, 1:20 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > Nope. The limit of the set of digits to the left of the decimal > > point is not a set of digits to the right of the decimal. > > Of course it is not, but it does not prohibit that there are digits on > the right. > > > If we change the limit to the set of digits to the left or right of > > the decimal point we still get {}. {} is not a real number > > and does not have a reciprocal. > > We cannot conclude from set theory that the digits on the right of the > decimal point vanish.
Yes we can.
> > > > > > Every infinite sequence of real numbers either has no limit or has a > > > limit in the real numbers or the improper limit oo. In any case there > > > are never two or more limits! > > > Piffle. You really know nothing about limits do you. > > In my book on analysis I write: a sequence may have many accumulation > points, If there is only one accumulation point, we call it the limit > of the sequence. (But I did not invent that definition.)
Anyone who is writing a book on analysis should understand that accumulation point depends on the topology used. Sure, one usually use the "standard" topology derived from the standard metric, and one may even use language suggesting that this is the only possible topology, but you still have to know there are other possibilities.
