We have already seen the axioms for Archimedean ordered fields and how such fields contain a copy of the rational numbers, . This web page discusses some beautiful and very powerful consequences of these facts all describing how the rationals are embedded in an Archimedean ordered field such as the reals.
All the results here concern an Archimedean ordered field and the copy of the set of rationals embedded in it. To help your intuition (and to spell out the most important case) we shall call our field but the results here apply to a general Archimedean ordered field too.
We start with a useful equivalent formulation of the Archimedean propery.
Let be an element of . Then there is such that .
Since , is defined and positive. Let be such that . (This exists by the Archimedean Property.) Then by the proposition on reciprocals in ordered fields.
The next result is rather beautiful and gets used a lot in more advanced work.
Density of the rationals in the reals.
Let with . Then there is some with .
Using the previous proposition, let satisfy . Let be the least natural number such that . (Note that with exists by the Archimedean Property and we may take the least such by the least number principle.) Then for if we have
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 is
closed under addition, multiplication, etc., it is not closed
under taking limits of rational sequences. (We will say that
the field is not
A sequence is monotonic nondecreasing if for all . It is monotonic nonincreasing if for all . It is monotonic if it is either monotonic nondecreasing or monotonic nonincreasing.
For any there is a monotonic nondecreasing sequence of rationals converging to .
We start by defining a sequence using induction on . Start with any in . (Such can actually be found in by the Archimedean Property.)
Inductively assume that is defined and . If let . Otherwise, let where is largest such that . Such exists by a combination of the Archimedean Property and the least number principle, as in the previous proof.
By construction, is a nondecreasing sequence of rationals and bounded above by . We show that it converges to .
Let be arbitrary. It will suffice to show that there is such that , for this means so for all by the fact that the sequence is nondecreasing. If not, we have that all . We show that this leads to a contradiction.
So, given that all , let be such that , by the Archimedean Property. Since we have . Thus in the construction, where is largest so that . But, by the assumption we are making, we also have so . This means contradicting the definition of , since it seems that is a larger natural number that would have worked. This is our contradiction and we conclude that some , as required.
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 ,
or indeed of any Archimidean ordered field. We've seen this
illustrated in three different ways: firstly
globally, using the relation to
explain what this
density means, and finally by showing that
every real number is the limit of a nondecreasing
sequence of rationals.