This section discusses the construction of the real numbers from
the rationals via the idea of a monotonic nondecreasing sequence.
This approach would appear to be perhaps the best one of all from
our point of view that the monotone convergence theorem
is the most natural formulation of the completeness axiom, however
it is not the only possible approach.
In fact, there is an important technical problem concerning this
choice of using monotonic nondecreasing sequences as a base for
the construction of the reals that makes it more difficult to work
with negative numbers directly. This is because the additive
inverse or negative
of a nondecreasing sequence is most naturally
a nonincreasing sequence. For this reason, this web page
starts with the construction of the nonnegative reals and
then gives the construction of the full real number line from
its nonnegative part at the end.
Other approaches to constructions of the reals are discussed
in other web pages in this section, as linked from here and from
the contents page.
This web page is still under construction. The final version
will contain some (but not all) of the proofs omitted here at present.
A bounded monotonic nondecreasing sequence of nonnegative rationals is
a sequence (_{
)
} such that: (a) each _{
} is
a nonnegative rational number; (b) _{
}_{
+1
}
for all
; and (c) there is a bound
such that _{
} for all
.
The set of nonnegative rationals will be denoted
^{+}
and the set of nondecreasing sequences of nonnegative rationals
will be denoted on this page by
^{+}
.
This set
^{+}
is not the set of (nonnegative) reals, but
closely related to it. For example,
^{+}
contains sequences
converging to what we think of as 2. The problem is
that
^{+}
contains several such sequences and we
want there to be only one 2. The solution
is to use an equivalence relation to make elements of
^{+}
the same.
The idea is that two monotonic nondecreasing sequences are equivalent
if each contains terms greater than terms in the other. In other words,
this will mean that they tend to the same real limit. (Notice that we
can't talk about this limit as we haven't constructed the real numbers
yet so we have to find a roundabout way of saying that they tend to the same real limit.)
Proposition.
The relation is an equivalence relation on
^{+}
. For each ,
, the constant sequences
with values , respectively are equivalent if and
only if =
.
Proof.
It is obviously symmetric because the definition of was
deliberately symmetric in (_{
),(
)
}.
If
_{
}
and
_{
}
then given arbitrary
there is
such that _{
} and
such that
_{
} hence
_{
}
.
The other property is proved similarly.
Definition.
Given a sequence (_{
)
+
} we write
_{
}
for the equivalence class of (_{
)
}.
We define
^{+}
to be the set of equivalence classes
_{
}
for (_{
)
+
}. For
^{+}
we identify with the constant sequence
with value . This means that we can think of
^{+}
as containing (a copy of)
^{+}
.
There is something to prove here, but it is easy: we must show that
two constant sequences with values ,
^{+}
are inequivalent if
. In other words, if
then the reals identified with and are different.
But this is easy as if
, _{
=
}, _{
=
}
then it is not true that
_{
}
.
We also have to define the order on , as
well as the usual operations of addition, multiplication, etc.
Definition.
We write
_{
}
to mean
_{
}
We write
_{
}
to mean
_{
}
; and we write
_{
}
to mean that
_{
}
and these
equivalence classes are not equal. We also define
_{
}
to mean
_{
}
.
In particular, we write
_{
0
}
to mean
0
_{
}
Proposition.
The order relation on the set
^{+}
is welldefined and satisfies the following.
Proof.
To make sure that this definition makes sense we should check that
if
_{
}
and
(
_{
)(
)
}
,
(
_{
)(
)
}
then
_{
}
.
This argument is the same as the argument for transitivity
of given above: use
_{
}
,
_{
}
,
_{
}
.
We also need to verify at this stage that this order relation on
^{+}
is compatible with the usual one on
^{+}
via the embedding defined
above. This is easy too: if
^{+}
and
_{
=
}, _{
=
} then it is easy to see that
_{
}
.
The other axioms are left as exercises for now.
Note that the last part of the above proposition is a version
of the Archimedean property for
^{+}
.
Definition.
The real number
_{
+
}
is defined
to be the equivalence class of the sequence
_{
}
defined by
_{
=
+
}
.
The real number
_{
}
is defined
to be the equivalence class of the sequence
_{
=
}
.
Proposition.
The addition and multiplication
operations +, just given are welldefined on
^{+}
.
In particular, the definitions above
do not depend on the choice of representative of equvalence class.
This defines the nonnegative reals,
^{+}
as a structure
extending
^{+}
with order, , and addition and
multiplication stucture. Because we have not defined  (indeed,
we cannot define this of
^{+}
) we are not able to prove
the field axioms in
^{+}
. But most
of the obvious algebraic properties are summed up in the next proposition.
Proposition.
The set
^{+}
with operations +,
and order relation satisfy, for all ,,
^{+}
,
 (+)+=+(+)

+=+

+0=
 (
)=(
)

=

0=0

0
1=

0
^{+}
=

(+)=
+

+
+


^{+}
+=
Although this only defines the nonnegative reals, this is enough
to make the idea of a sequence of nonnegative real numbers to make sense,
and in particular to interpret the idea of limit in
^{+}
.
When we do this we get,
Proposition.
The set
^{+}
has the completeness property,
that every monotonic nondecreasing bounded sequence
(_{
)
} of numbers _{
+
}
has a limit
^{+}
.
To pass from the nonnegative reals to the full set of reals,
one just needs to repeat a construction given earlier when we
constructed the from
the
.
Definition.
Let be the following relation defined on
^{+
+
}
: we define
(
_{1,
1)(
2,
2)
}
to mean
_{1+
2=
2+
1
}
.
Proposition.
is an equivalence relation on
^{+
+
}
.
Definition.
We write
,
for
the equivalence class of
(,)
^{+
+
}
.
The set is the set
^{+
+/
}
of equivalence classes.
Proposition.
The function defined by
()=
,0
is a onetoone function mapping
^{+}
into .
Proof.
Let ,
^{+}
with ()=()
.
Then
,0=
,0
hence
(,0)(,0)
hence +0=+0
hence
=
.
The elements of
^{+}
are identified with
their images ()
in the usual way.
Definition.
We define the arithmetic structure on reals as follows.

_{1,
2+
1,
2
}
=
_{1+
1,
2+
2
}

_{1,
2
1,
2
}
=
_{1+
2,
2+
1
}

_{1,
2
1,
2
}
=
_{1
1+
2
2,
2
1+
1
2
}

_{1,
2
1,
2
}
_{1+
2
2+
1
}
Theorem.
These definitions are welldefined. They make
into an Archimedean ordered field satisfying the completeness axiom.