Whether an ordered field $K$ equipped with the Nested Interval Property or The Cauchy Complete Property is an archimedean ordered field [closed]

Question

The Nested Interval Property

For all sequences $\{a_n\},\{b_n\}\in K^\mathbb N$ satisfied that for every element $\varepsilon\in K_{>0}$ , there exists an $N\in\mathbb N$ such shat $|a_n-b_n|<\varepsilon$ for all $n>N$ and$$a_0\le a_1<\cdots\le a_n<\cdots\le b_n\le \cdots\le b_1\le b_0$$then there is $c\in K$ such that $$\bigcap_n[a_n,b_n]=\{c\}$$

The Cauchy Complete Property

For all $\{a_n\}\in K^\mathbb N$ satisfied forall $\varepsilon\in K_{>0}$, there is a $N\in \mathbb N$ such that $|a_n-a_m|<\varepsilon$ for all $n,m>N$ , then there is a $c\in K$ such that for all $\varepsilon\in K_{>0}$ there is a $N\in \mathbb N$ such that $|a_n-c|<\varepsilon$ for all $n>N$

The Archimedean Property

For all $x\in K$ and $y\in K_{>0}$ , there is $n\in\mathbb N$ such that $n\cdot y>x$ .

$n\cdot y$ is $$\underbrace{y+y+\cdots+y}_{n \ \text{times}}$$

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I know an archimedean ordered field equipped with the Nested Interval Property or the Cauchy Complete Property is isomorphic to $\mathbb R$ and I'm wandering whether it can be proved except the Archimedean Property.

Answer 1

No, the Archimedian property is necessary. The ease of constructing a counterexample depends on your mathematical background.

Before I begin: (Cauchy) completeness is easily shown to be equivalent to the Nested Interval Property. I will refer to the two interchangeably as completeness.

At the level of intuition, the problem is that the decimal expansion of a real number is "only" as long as $\mathbb{N}$. There are analogues with longer "decimal sequences"^Fn. 1, called the surreal numbers. Wikipedia's (linked) construction is quite good, and no more complicated than Rudin's construction of $\mathbb{R}$ in Principles of Mathematical Analysis.

The Archimedian property encodes the length of the decimals when it restricts $n\in\mathbb{N}$. Your definition of completeness^Fn. 2 does not; it depends on $\epsilon\in K^+$, which may be too small for decimals. It admits more examples than the Archimedian property: the simplest is the Levi-Civita field; the most complex (IMHO) is the aforementioned surreals.

If you've seen some mathematical logic, then the nonstandard reals ${}^*\mathbb{R}$ are a slick counterexample. That ${}^*\mathbb{R}$ is non-Archimedean is well-known: for any nonprincipal ultrafilter $\mathcal{U}$ and any $r\in\mathbb{R}$, $$(0,1,2,3,\dots)/\mathcal{U}>(r,r,r,\dots)/\mathcal{U}$$ But Cauchy Completeness is easily formalizable in first-order logic with quantification over $\mathbb{N}$, and so holds in ${}^*\mathbb{R}$ by the transfer principle. (The transfer principle shows completeness for sequences indexed by ${}^*\mathbb{N}$, but this implies standard completeness a fortiori.)

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^Fn. 1 Technically, Dedekind cuts.

^Fn. 2 There are other definitions of completeness that "bake in" the reals; these are equivalent to Archimediacy.