Can the monotone convergence theorem (every bounded monotone sequence of real numbers has a limit) be...

I recently gave my honors real analysis students the assignment of
deriving the monotone convergence theorem (every bounded monotone
sequence of real numbers has a limit) from the Cauchy convergence
theorem (every Cauchy sequence of real numbers has a limit), using as
background only the ordered field properties of R -- i.e., without
using the least upper bound property, the nested intervals property,
the Bolzano-Weierstrass theorem, or any of the other (equivalent)
completeness properties of R.  After all, every working mathematician
"knows" that all the analytic completeness properties of R are
deducible from one another.

Well, it turns out that I gave them an impossible assignment: No such
derivation is possible!  Here's why:  Let F be the field of Laurent
series in x with coefficients in Q (or in R, if you prefer).  F has a
natural ordering that makes it an ordered field (we call a Laurent
series "positive" if its leading coefficient is).  A sequence

        p (x), p (x), ...
         1      2

of Laurent series will be called Cauchy if for all Laurent series
q(x) > 0, there exists K such that for all m,n > K,

        - q(x) < p (x) - p (x) < q(x) .
                  m       n

(NB: This is the definition of "Cauchy" appropriate to ordered fields,
and does not coincide with the definition of "Cauchy" used for metric
spaces.  For the proof-theoretic purposes at hand, it's the right
definition to use.)  It is not hard to show that a sequence of Laurent
series is Cauchy if and only if every coefficient eventually
stabilizes.  But this implies that the sequence is convergent.  Hence
every Cauchy sequence in F converges.  At the same time, F fails to
satisfy the supremum property: for instance, the terminating Laurent
series

        p (x) = n
         n

(n = 1, 2, ...) have q(x) = 1/x as an upper bound, but have no least
upper bound.  Thus, we have an ordered field that satisfies the Cauchy
completeness property but not the least-upper-bound completeness
property.  And this, of course, tells us that the assignment I gave
was impossible.

None of this is especially deep, but it's the sort of thing people
should know (especially if they plan to teach courses in real analysis
and wish to avoid assigning impossible problems!), so I've thought
about writing a Note in the Monthly about it.  But before I do, I'd
like to make sure there's a real need for such a Note.  Has anyone out
there seen this fact about the Cauchy convergence property discussed
in any of the major non-research journals in the last twenty or so
years?

I'd also be interested in knowing about textbooks on real analysis
that treat this issue correctly (if any exist), or, failing that,
research articles or monographs that go into the minutiae of what
theorems in real analysis imply what other theorems.

-- Jim Propp <propp@cartan.berkeley.edu>