Re: Dedekind Cuts, Fundamental Sequences: why?

On Wed, 06 Jun 2007 22:25:46 -0400, Hatto von Aquitanien
<abbot@xxxxxxxxxxxxxx> wrote:

Dave Seaman wrote:

On Wed, 06 Jun 2007 15:30:43 -0400, Hatto von Aquitanien wrote:
David C. Ullrich wrote:

The rationals are not complete -

By what criteria of completeness? I have presented two possible ways of
viewing the rational numbers as being incomplete, and hinted at a third.

Completeness of the reals means that the least upper bound axiom is
specified. Equivalently, that every Cauchy sequence converges.

The latter seems to be a tautology.

<quote>

A Cauchy sequence of rational numbers is a sequence x of rational numbers
such that for every positive rational number epsilon there exists a
positive integer N such that for every m, n > N

\ | x_n - x_m | < epsilon

</quote>

That's not a tautology, it's a _definition_.

Looks like you're done. What do you need real numbers for?

To get a structure where every Cauchy sequence is _convergent_!
The definition of "convergent" is this:

"The sequence (x_n) is convergent if there exists x such that
for every epsilon > 0 there exists N such that for every n > N

|x - x_n| < epsilon."

If you think that a Cauchy sequence is automatically convergent
you're simply wrong.



we want to extend them to a complete ordered field, so that we can prove
the basic theorems of analysis.

That statement merely begs the question. Quite honestly, my senior level
real analysis course bored the piss out of me because most of it was
blatantly obvious.

Then you must find the study of incomplete ordered fields (such as the
rationals) to be even more obvious. So all your objections apply to that
case in spades.

I don't know what that would necessarily follow. Most of real analysis
appeals to the intuitive notions of continuity. It's a very different way
of thinking from what is used in discrete math.


************************

David C. Ullrich
.

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