As I've mentioned elsewhere,
note that the two standard constructions of the reals
(Dedekind and Cauchy)
correspond to two presentations of the topology on $\bR$,
namely by intervals and by balls,
and to the two natural generalizations of $\bR$,
namely -lines- (er, linear continua)
and -metric spaces-.
The latter is a more useful generalization, hence why we prefer Cauchy
sequences.

Also, note that Cauchy sequences are, IMHO, -easier- than Dedekind
cuts.

That is, there's a -strong- analogy between the construction of $\bQ$
as -equivalence classes- of pairs
and $\bR$ as -equivalence classes- of cauchy sequences -- while
Dedekind cuts come out of the blue.

Indeed, every schoolchild is (or should be) familiar with the fact
that $0.9\dots = 1.0\dots$, IE, that decimal representations are not
unique -- so it's only a small step to seeing that we can use -all-
Cauchy sequences.
