The Cauchy property
is a useful idea that
describes sequences that seem to converge
without mentioning any limit. It is a modification of the usual
definition of convergence except that we cannot compare the values of
the sequence to
Definition.
We say that a sequence (_{ ) } has the Cauchy property if the following statement holds:
Convergent sequences do have the Cauchy property. This is quite easy to prove.
Proof.
We're going to prove the converse to this result now, that every Cauchy sequence (i.e., one with the Cauchy property) converges to some limit. Once again, the completeness of the real numbers is the fact that makes this work. First we need a useful lemma, the proof of which is almost identical to the theorem that says every convergent sequence is bounded.
Proof.
Proof.
By the lemma, (_{
)
} is bounded, so by the Bolzano-Weierstrass theorem
(which used the completeness of the reals) there is
Let
as required.
The real importance of this result is that it enables us to state the completeness
axiom for the reals in a way that uses the distance function only, and not
using the order relation on the reals. Indeed many people prefer to take
as their completeness axiom the statement Every Cauchy sequence has a limit.
(Note that the monotone convergence theorem needs the order relation
The main result just presented (that every Cauchy sequence has a limit)
is another version of the completeness property for the fields.
Because it doesn't require the order relation,
Completeness, Cauchy form.
An example of a field with a distance function which isn't ordered is the field
of complex numbers,
The two approaches to completeness for Archimedean ordered fields
such as the reals (via monotonic sequences and via Cauchy sequences)
are equivalent: it is possible to show directly from the Archimedean
Property of the reals that every bounded monotonic sequence is Cauchy,
and hence the above theorem implies that every bounded monotonic
sequence has a limit. So for the reals it is entirely a matter of
choice (or taste) which approach one prefers. Personally, I find the
monotone convergence theorem more obviously true
and therefore
preferable as an axiom for the reals. However, as mentioned
already, there are other situations in which the Cauchy sequence
approach is the only one possible.