# every ordered field with the least upper bound property is isomorphic to $\mathbb{R}$, proof that

Let $F$ be an ordered field with the least upper bound property. By the
order properties of $F$, $$ and by an induction^{} argument^{} $$
for any positive integer $n$. Hence the characteristic^{} of the field $F$ is
zero, implying that there is an order-preserving embedding^{} $j:\mathbb{Q}\to F$.

We would like to extend this map to an embedding of $\mathbb{R}$ into $F$. Let $r\in \mathbb{R}$ and let $$ be the associated Dedekind cut. Since ${D}_{r}$ is nonempty and bounded above in $\mathbb{Q}$, it follows that the set $j({D}_{r})$ is nonempty and bounded above in $F$. Applying the least upper bound property of $F$, define a function $\stackrel{~}{\u0237}:\mathbb{R}\to F$ by

$$\stackrel{~}{\u0237}(r)=sup\left(j({D}_{r})\right).$$ |

One can check that $\stackrel{~}{\u0237}$ is an order-preserving field homomorphism.
By replacing $F$ with the isomorphic^{} field $F\setminus \stackrel{~}{\u0237}(\mathbb{R})\cup \mathbb{R}$,
we may assume that $\mathbb{R}\subset F$.

We claim that in fact $\mathbb{R}=F$. To see this, first recall that since $F$ is a partially ordered group with the least upper bound property, $F$ has the Archimedean property (http://planetmath.org/DistributivityInPoGroups). So for any $f\in F$, there exists some positive integer $n$ such that $$. Hence the set $$ is nonempty and bounded above, implying that ${f}^{\prime}={sup}_{\mathbb{R}}{D}_{f}^{\prime}$ lies in $\mathbb{R}$. Now observe that applying the least upper bound axiom in $F$ gives us that $f={sup}_{F}{D}_{f}^{\prime}$. Since ${f}^{\prime}$ is an upper bound of ${D}_{f}^{\prime}$ in $F$, it follows that $f\le {f}^{\prime}$.

Seeking a contradiction^{}, suppose $$. By the Archimedean property,
there is some positive integer $n$ such that $$. Because
${f}^{\prime}={sup}_{\mathbb{R}}{D}_{f}^{\prime}$, we obtain $$, which implies the
contradiction
$$. Therefore $f={f}^{\prime}$, and so $f\in \mathbb{R}$. This completes^{} the proof.

Title | every ordered field with the least upper bound property is isomorphic to $\mathbb{R}$, proof that |
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Canonical name | EveryOrderedFieldWithTheLeastUpperBoundPropertyIsIsomorphicTomathbbRProofThat |

Date of creation | 2013-03-22 14:10:51 |

Last modified on | 2013-03-22 14:10:51 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 9 |

Author | mps (409) |

Entry type | Proof |

Classification | msc 12E99 |

Classification | msc 54C30 |

Classification | msc 26-00 |

Classification | msc 12D99 |