Re: Dedekind Cuts, Fundamental Sequences: why?

Dave Seaman wrote:

On Fri, 08 Jun 2007 21:58:04 -0400, Hatto von Aquitanien wrote:

Actually, what you said is this:

"The number L that appears in the definition is called the limit of the
sequence.  I have been assuming that you were familiar with the
definitions, since you claimed to have taken a senior-level real analysis
course.  (Senior level in what, I might ask.  High School?)"

That's what immediately preceded the paragraph that I then quoted. I
actually said both those things.

But the paragraph you quoted here supports my claim that a Cauchy
sequence of rationals does not necessarily converge, because in some
cases there is no rational L that fits the definition.


And then there's: "I don't care whether you agree or not.  'Convergence'
implies the existence of a limit."

Other than the limit abs(a_m-a_n) -> 0 as N -> infinity where m,n>N, yours
is not a consensus view. A Cauchy sequence {a_n} converges. (The period
indicates the end of the sentence.) Whether it "converges to a limit in
the domain of its elements" is a different matter. I have cited multiple
authoritative source on this, and I can continue to provide more.

"Cauchy's Criterion for Convergence. In accordance with Cantor's basic
idea, real numbers can be described by convergent rational sequences. Two
rational sequences (r_n) and (s_n) have the same (real) limit, if and only
if the sequence of their differences (r_n-s_n) converges to zero. It is
natural therefore to _define_ the _real_ _numbers_ as equivalence classes
of _convergent_ _rational_ _sequences_ ; two sequences being equivalent
when their difference sequence converges to zero. _For_ _this_
_definition_ to be meaningful, the _convergence_ of a sequence has to be
_characterized_ _without_ making _use_ of _limits_ . This can be done with
the help of Cauchy's criterion, which will be used to define the sequences
concerned."

http://www.amazon.com/Numbers-Graduate-Texts-Mathematics-Readings/dp/0387974970

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