Re: Dedekind Cuts, Fundamental Sequences: why?

"Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
news:f4e76p$ti3$2@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Sat, 09 Jun 2007 01:34:05 -0400, Hatto von Aquitanien wrote:

"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

Although the consensus in sci.math seems to be otherwise, the way I
actually learned it in my graduate real analysis course is that we con't
speak of "Cauchy sequences of rationals" when constructing the real
numbers. We speak of "fundamental sequences of rationals" instead.
I believe this is Cantor's terminology.

[...]

Yes, I see your point. At my institution the usual technique is to talk
about general sequences in a topological space, and then specialise the
discussion to infinite sequences in Q. We can talk of convergence of
subsequences of these infinite sequences with only topological notions. The
completion follows.

Once this process is completed, we teach the students Cauchy sequences and
their relevance, to much whining and gnashing of teeth.

I do not think this is so bad, as it allows us to talk about many
interesting related theorems in topology.

--
Glen


.

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