We have already seen the axioms for
Archimedean ordered fields and
how such fields contain a copy of the rational numbers,
All the results here concern an
Archimedean ordered field
and the copy
We start with a useful equivalent formulation of the Archimedean propery.
The next result is rather beautiful and gets used a lot in more advanced work.
Density of the rationals in the reals.
Using the previous proposition, let
By a similar sort of argument one can prove much more than this:
that every real number is the limit of a sequence of rational
numbers. This too is a powerful result that is discussed in other
topics and courses in analysis. It also shows something important
about the rational numbers, that although
Inductively assume that
By construction, (
So, given that all
Either by modifying the above proof, or by using the result it shows, show that every real number is the limit of a nonincreasing sequence of rationals.
The rationals form a
dense subset of
globally, using the
density means, and finally by showing that
every real number is the limit of a nondecreasing
sequence of rationals.