The five completeness properties of an ordered field: least upper bound, monotone convergence, nested intervals, Bolzano-Weierstrass, and Cauchy completeness

Throughout, $F$ is an ordered field (Ordered field) with its order and its absolute value. Sequences in $F$, and the notions of convergence in $F$, Cauchyness in $F$, boundedness, nondecreasing and nonincreasing, subsequence, closed interval $[a,b]_F$, nesting, and lengths tending to $0$ in $F$, are the ones fixed once and for all in Sequences, convergence, Cauchyness, monotonicity, boundedness and closed intervals in an arbitrary ordered field. They are not restated here and they are never read in $\mathbb{R}$: every $\varepsilon$ below ranges over the positive elements of $F$ itself.

A sequence $(x_k)$ in $F$ is bounded above when there is $B \in F$ with $x_k \le B$ for every $k \in \mathbb{N}$, and a subset $S \subseteq F$ is bounded above when there is $B \in F$ with $s \le B$ for every $s \in S$ (Complete ordered field (least-upper-bound property), Upper bound, least upper bound, and strict upper bound).

The following are five properties that $F$ may or may not have.

• (LUB), the least-upper-bound property. Every nonempty $S \subseteq F$ that is bounded above has a least upper bound in $F$. This is exactly the condition that makes $F$ a complete ordered field (Complete ordered field (least-upper-bound property)), and the two names are used interchangeably here.

• (MCT), the monotone convergence property. Every nondecreasing sequence in $F$ that is bounded above converges in $F$.

• (NIP), the nested interval property. For every nested sequence $(I_k)_{k \in \mathbb{N}}$ of closed intervals $I_k = [a_k, b_k]_F$ of $F$ whose lengths tend to $0$ in $F$, the intersection

  $\bigcap_{k \in \mathbb{N}} I_k$

  is nonempty.

• (BW), the Bolzano-Weierstrass property. Every bounded sequence in $F$ has a subsequence that converges in $F$.

• (CC), Cauchy completeness. Every Cauchy sequence in $F$ converges in $F$.

Alongside these we use the Archimedean property (ARCH) of Archimedean ordered field: for every $x \in F$ there is a natural number $n$ with $x < n \cdot 1_F$.

• (NIP) is stated in the shrinking form because that is the form both satisfied by the formal Laurent series field and used by the bisection theorem. The field $\mathbb{R}((t^{-1}))$ is Cauchy complete without having least upper bounds (Every Cauchy sequence in $\mathbb{R}((t^{-1}))$ converges: $K$ is sequentially Cauchy complete, $\mathbb{R}((t^{-1}))$ does not have the least-upper-bound property; its canonical naturals have no supremum), and it satisfies shrinking (NIP) ($\mathbb{R}((t^{-1}))$ has the nested interval property for lengths tending to $0$). The same shrinking condition is exactly what the bisection argument of Nested intervals plus the Archimedean property imply Bolzano-Weierstrass, by repeated bisection produces.

• "Lengths tend to $0$" is read in $F$. For a non-Archimedean $F$ this is strictly stronger than the same words read in $\mathbb{R}$ through some identification of the rational scalars, and the difference is not academic: the remarks of $\mathbb{R}((t^{-1}))$ has the nested interval property for lengths tending to $0$ exhibit intervals in $\mathbb{R}((t^{-1}))$ whose lengths are the real constants $2/(n+1)$, which tend to $0$ in the ordinary real sense and do not tend to $0$ in the order of that field.

• Boundedness of a sequence is two-sided, boundedness above is not. Sequences, convergence, Cauchyness, monotonicity, boundedness and closed intervals in an arbitrary ordered field calls $(x_k)$ bounded when $|x_k| \le M$ for every $k$, which is the hypothesis of (BW); (MCT) asks only for the one-sided bound $x_k \le B$, which for a nondecreasing sequence is the only side in question, since $x_0 \le x_k$ always.

• (MCT) is stated for nondecreasing sequences only. The nonincreasing case is not a separate assumption: if $(x_k)$ is nonincreasing and bounded below by $b$ then $(-x_k)$ is nondecreasing and bounded above by $-b$, and $x_k \to L$ exactly when $-x_k \to -L$, because $|(-x_k) - (-L)| = |x_k - L|$. That reduction is used in the proof of The monotone convergence property plus the Archimedean property imply the least-upper-bound property.

• Nothing here presumes that any of the five holds. They are predicates on an ordered field, and the point of the page they open is that in the presence of (ARCH) they are all the same predicate (For an ordered field the five completeness properties are equivalent, provided the Archimedean property is assumed alongside nested intervals and Cauchy completeness), while without it two of them are strictly weaker (FALSE: the nested interval property alone implies the least-upper-bound property, FALSE: an ordered field in which every Cauchy sequence converges has the least-upper-bound property).

• (CC) is this library's third rendering of "Cauchy complete", and for $\mathbb{R}$ all three agree. Cauchy sequence in a metric space and Complete metric space: every Cauchy sequence converges in the space read Cauchyness and completeness in a metric space, and the case $n = 1$ of $\mathbb{R}$ and $\mathbb{R}^n$ for $n \ge 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $\mathbb{R}$ proves $(\mathbb{R}, |x-y|)$ complete; Limits and Cauchy sequences of reals reads both notions for real sequences, with $\varepsilon$ ranging over the positive rationals; the present definition reads them in an ordered field $F$. For $F = \mathbb{R}$ under the metric $d(x,y) = |x-y|$ of The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded the three unfold to the same quantified statement: below every positive real lies a positive rational (The rationals embed densely in the reals), so the two ranges of $\varepsilon$ pick out the same Cauchy sequences and the same convergent ones. So "$\mathbb{R}$ satisfies (CC)" is a statement this library has already proved twice, as The Cauchy criterion from the least-upper-bound property: in a complete ordered field every Cauchy sequence converges and as the case $n = 1$ of $\mathbb{R}$ and $\mathbb{R}^n$ for $n \ge 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $\mathbb{R}$. The parallel stops at $\mathbb{R}$. The absolute value of an ordered field takes its values in $F$, while a metric is required to take its values in $\mathbb{R}$ (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric), so for a non-Archimedean $F$ the map $(x,y) \mapsto |x-y|$ is not a metric in this library's sense and the metric development says nothing about it. That is why Sequence basics in an arbitrary ordered field: limits are unique, limits preserve non-strict inequalities, convergent sequences are Cauchy, Cauchy sequences are bounded, and a Cauchy sequence with a convergent subsequence converges had to be proved from the order axioms alone, although its Cauchy clauses reappear for metric spaces as Every convergent sequence in a metric space is Cauchy, Every Cauchy sequence in a metric space is bounded and A Cauchy sequence in a metric space with a convergent subsequence converges to that subsequence’s limit. Neither development generalises the other; the comparison above identifies their agreement at $\mathbb{R}$ without claiming that $\mathbb{R}$ is their only overlap.