This section discusses the construction of the real numbers from the rationals via the idea of a Cauchy sequence.
This approach is perhaps the most natural one from the point of
view of Cauchy's version of the completeness axiom,
and has the advantage that it generalizes to other settings where there is
a distance function
+
,
This web page is still under construction. The final version will contain some (but not all) of the proofs omitted here at present.
We start by recalling the definition of a Cauchy sequence.
Definition.
A sequence (_{
)
} of rationals is said to be Cauchy
(or: has the Cauchy Property) if
with the equivalent
.
(This is equivalent because of the Archimedean property, of course.)
We have seen that we expect
all Cauchy
sequences to converge, and all reals
to be the limit of a Cauchy sequence of rationals.
The Cauchy property is particularly useful as it
doesn't mention the limit
Definition.
We let
Once again, a real number may have more than one rational Cauchy sequence converging to it, so we must factor out by an equivalence relation to obtain the true version of the reals.
Definition.
For Cauchy sequences of rationals
(_{
),(
)
},
we say (_{
)(
)
} if
Proposition.
The relation
Proof.
For a rational,
, the constant sequence with value
is a Cauchy sequence. You can check that two distinct constant
sequences _{1}
and _{2}
are inequivalent. Therefore
we can identify each
with the equivalence class
We expect the arithmetic operations of +,
Definition.
This completes the definition part, as we have now defined
Theorem.
The set
Proof.
Another long exercise (sigh).