Re: .999... = 1

David C. Ullrich wrote:
On Mon, 29 Aug 2005 16:02:59 +0100, Robert Low <mtx014@xxxxxxxxxxxxxx> 
That's pretty much it: you can think of a real number as
precisely being an equivalence class of Cauchy sequences
of rationals.

However, to me, .99999... represents a sequence that tends to 1,
rather than 1.
OK, so it's a sequence of rationals that tends to 1. 
No, 0.999... is most certainly not a sequence of rationals.
It is the limit of a certain sequence of rationals.


Eventually it is. But surely 0.999... means 'the real number
defined by the Cauchy sequence 0.9, 0.99, 0.999 ...', since we
have to make sense of the infinite sum. And reals are defined
as equivalence classes of Cauchy sequences of rationals, so
we have a Cauchy sequence 0.9, 0.99, 0.999 etc which lies
in the same equivalence class as the Cauchy sequence 1,1,1 etc,
so they represent the same real.

Unless, of course, you think of the reals as the unique
complete ordered field...
.

Relevant Pages

  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... What is the step of logic which leads one to seek an extention of the ... But when they go from the rationals to the reals, ... class of Cauchy sequences that converge to that same point. ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... What is the step of logic which leads one to seek an extention of the ... But when they go from the rationals to the reals, ... class of Cauchy sequences that converge to that same point. ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... using the rationals as a starting point. ... that one could mix rational numbers with 'irrational Cauchy sequences', ... Do you mean each equivalence class is ... The top layer, so far, builds the reals from the rationals. ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... using the rationals as a starting point. ... that one could mix rational numbers with 'irrational Cauchy sequences', ... or do you mean that the set of equivalence classes is ... the reals strictly from the rationals. ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... What is the step of logic which leads one to seek an extention of the ...  But when they go from the rationals to the reals, ... class of Cauchy sequences that converge to that same point. ... built up, step by step, from rationals, to sequences ...
    (sci.math)