I want to show that if an ordered field $X$ has the least upper bound property (meaning, every nonempty set $E$ which is bounded above has $\sup E \in X$), then it is complete (meaning, every Cauchy sequence converges in $X$).

I know the converse is not true, but how do I prove this direction?

  • $\begingroup$ I believe it is not the case that the converse is not true, but rather that it is not strong enough. For example we may need the Archimedean property to prove the completeness axiom. If however you start with the monotone convergence theorem you can show that it implies the completeness axiom $\endgroup$ – user38268 Mar 18 '12 at 0:50
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    $\begingroup$ Its not really clear what you mean by "bounded above" or "sup E." These aren't properties of subsets of metric spaces, generally. $\endgroup$ – Tim kinsella Mar 18 '12 at 0:51
  • $\begingroup$ Is your metric space ordered somehow? How does the order relate to the metric? This doesn't really make a lot of sense to me as written... $\endgroup$ – Arturo Magidin Mar 18 '12 at 0:56
  • $\begingroup$ @ArturoMagidin Whoops I tacitly assumed the OP was talking about $\mathbb{R}$...maybe that is what the OP means? $\endgroup$ – user38268 Mar 18 '12 at 1:26
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    $\begingroup$ Sorry I meant ordered field! $\endgroup$ – jamaicanworm Mar 18 '12 at 1:30

In an arbitrary ordered field one has the notion of Dedekind completeness -- every nonempty bounded above subset has a least upper bound -- and also the notion of sequential completeness -- every Cauchy sequence converges.

The main theorem here is as follows:

For an ordered field $F$, the following are equivalent:
(i) $F$ is Dedekind complete.
(ii) $F$ is sequentially complete and Archimedean.

Since the Archimedean hypothesis often goes almost without saying in calculus / analysis courses, many otherwise learned people are unaware that there are non-Archimedean sequentially complete fields. In fact there are rather a lot of them, and they can differ quite a lot in their behavior: e.g. some of them are first countable in the induced (order) topology, and some of them are not. (In particular there are some ordered fields in which a sequence is Cauchy if and only if it is eventually constant! This is a case where one should consider Cauchy nets if one is serious about exploring the topology...)

These issues are treated in $\S 1.7$ and $1.8$ of these notes.

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    $\begingroup$ The note seems to be extremly helpful to me. Thanks! $\endgroup$ – Taxxi Dec 3 '14 at 10:43

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