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.
Proposition.
Let
Proof.
Since
The next result is rather beautiful and gets used a lot in more advanced work.
Density of the rationals in the reals.
Let
with
Proof.
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 complete
.)
Definition.
Theorem.
For any
Proof.
We start by defining a sequence (_{
)
}
using induction on
Inductively assume that _{
} is defined and _{
}.
If _{
+1
} let
_{
+1=}_{
}. Otherwise, let
_{
+1=}_{
+
} where
By construction, (_{
)
} is a nondecreasing sequence of rationals
and bounded above by
Let
So, given that all _{

}, let
Exercise.
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 near 0
,
then globally
, using the density
means, and finally by showing that
every real number is the limit of a nondecreasing
sequence of rationals.