Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema

In the setting of The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals, let $F$ be an ordered field in which every nonempty subset of $F$ that is bounded above has a supremum, and let $\kappa_{F}:\mathbb{Q}\to F$ be the canonical embedding.

¶ Every nonempty subset of $F$ that is bounded below has an infimum.

Let $x,y\in F$.

¶ There is $n\in\mathbb{N}$ with $x<n$, the natural number $n$ being read in $F$ as in Commutative Rings, Fields and Ordered Fields: Standard Notation §numerals.

¶ If $x<y$, there is $u\in\mathbb{Q}$ with $x<\kappa_{F}(u)<y$.

¶ $x$ is the supremum of $\kappa_{F}(\{u\in\mathbb{Q}:\kappa_{F}(u)<x\})$.