Sequences, convergence, Cauchyness, monotonicity, boundedness and closed intervals in an arbitrary ordered field

Throughout, $F$ is an ordered field (Ordered field) with its order and its absolute value $|\cdot|$ (Absolute value in an ordered field), and $\mathbb{N}$ is the set of natural numbers with its order (The natural numbers $\mathbb{N}$ (von Neumann), Order on the natural numbers).

A sequence in $F$ is a function $x : \mathbb{N} \to F$. We write $x_k$ for $x(k)$ and $(x_k)$, or $(x_k)_{k \in \mathbb{N}}$, for the function itself.

Let $(x_k)$ be a sequence in $F$.

• $(x_k)$ is bounded when there is $M \in F$ with $|x_k| \le M$ for every $k \in \mathbb{N}$.

• $(x_k)$ converges to $L \in F$ when

  $\text{for every } \varepsilon \in F \text{ with } \varepsilon > 0 \text{ there is } N \in \mathbb{N} \text{ such that } |x_k - L| < \varepsilon \text{ for all } k \ge N.$

  We then write $x_k \to L$ in $F$. The sequence is convergent in $F$ when it converges to some $L \in F$, and divergent in $F$ otherwise.

• $(x_k)$ is Cauchy in $F$ when

  $\text{for every } \varepsilon \in F \text{ with } \varepsilon > 0 \text{ there is } N \in \mathbb{N} \text{ such that } |x_k - x_l| < \varepsilon \text{ for all } k, l \ge N.$

• $(x_k)$ is nondecreasing when $x_j \le x_k$ for all $j \le k$, increasing when $x_j < x_k$ for all $j < k$, nonincreasing when $x_j \ge x_k$ for all $j \le k$, decreasing when $x_j > x_k$ for all $j < k$, and monotone when it is nondecreasing or nonincreasing.

• For a strictly increasing $n : \mathbb{N} \to \mathbb{N}$, the subsequence of $(x_k)$ along $n$ is the composite $(x_{n_j})_{j \in \mathbb{N}}$. An element $L \in F$ is a subsequential limit of $(x_k)$ when some subsequence of $(x_k)$ converges to $L$ in $F$.

Closed intervals and nesting. For $a, b \in F$ with $a \le b$, the closed interval with endpoints $a$ and $b$ is

$[a,b]_F \;:=\; \{\, x \in F : a \le x \le b \,\},$

and its length is $b - a \ge 0$. A sequence $(I_k)_{k \in \mathbb{N}}$ of closed intervals $I_k = [a_k, b_k]_F$ is nested when $I_{k+1} \subseteq I_k$ for every $k$. Its lengths tend to $0$ in $F$ when the sequence $(b_k - a_k)_{k \in \mathbb{N}}$ converges to $0$ in the sense above, that is, when for every $\varepsilon > 0$ in $F$ there is $N \in \mathbb{N}$ with $b_k - a_k < \varepsilon$ for all $k \ge N$ (the absolute value may be dropped because each length is $\ge 0$).

• The thresholds range over $F$, and that is not a stylistic choice. In an Archimedean ordered field one may equivalently test $\varepsilon$ over the canonical rationals, and that is what the $\mathbb{R}$-specific Limits and Cauchy sequences of reals does; the two agree there, as the remark on rational and real $\varepsilon$ in Sequences of reals: bounded, eventually, frequently, tails, subsequences records. In a general $F$ they do not agree, because the canonical rationals need not be cofinal below the positive elements. A concrete failure lives on this page: in $\mathbb{R}((t^{-1}))$ every positive rational constant exceeds $t^{-1}$ (clause 4 of $\mathbb{R}((t^{-1}))$ is non-Archimedean, and the monomials $t^{-k}$ are cofinal below its positive elements, since a nonzero constant is nonzero at index $0$), so the sequence taking the value $0$ at even indices and $t^{-1}$ at odd indices would satisfy the Cauchy condition read with rational thresholds only, while failing it at $\varepsilon = t^{-2}$, where consecutive terms differ by $t^{-1} > t^{-2}$; and it has no limit at all, since a convergent sequence is Cauchy by the triangle inequality. Every definition above therefore quantifies over $\varepsilon \in F$, and no proof in this library may substitute a rational threshold in a field that has not been shown to be Archimedean.

• These are the $\mathbb{R}$-notions with $\mathbb{R}$ replaced by $F$, and nothing more. Sequence, tail, subsequence and boundedness are Sequences of reals: bounded, eventually, frequently, tails, subsequences; monotonicity is Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences; subsequential limits are Subsequential limit of a real sequence, and the subsequential limit set; convergence and Cauchyness are Limits and Cauchy sequences of reals; closed intervals are the form $[a,b]$ of Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length. Only the field in which the inequalities are read has changed.

• Transfer of theorems is not automatic, and citing an $\mathbb{R}$-item for a general $F$ is a citation error. A result proved about sequences of reals is a statement about $\mathbb{R}$. Many such proofs use only the ordered-field axioms and go through for any $F$ verbatim, and many others use completeness or the Archimedean property and do not. Which is which has to be settled by reading the proof; until an item is stated for a general ordered field, it may not be cited for one.

• Limits are unique in any ordered field. If $x_k \to L$ and $x_k \to L'$ in $F$ with $L \ne L'$, put $\varepsilon := |L - L'|/2$, which is positive because $|L-L'| > 0$ (Basic properties of the absolute value) and $2 = 1 + 1 > 0$. Choose $N$ beyond which both $|x_k - L| < \varepsilon$ and $|x_k - L'| < \varepsilon$ hold, and take any $k \ge N$: the triangle inequality (The triangle inequality, proved for an arbitrary ordered field) gives $|L - L'| \le |L - x_k| + |x_k - L'| < 2\varepsilon = |L - L'|$, which is impossible. So the limit, when it exists, is unique, and the notation $\lim_k x_k$ is unambiguous. No completeness and no Archimedean hypothesis is used.

• Indexing starts at $0$, as everywhere in this library, because $0 \in \mathbb{N}$ (The natural numbers $\mathbb{N}$ (von Neumann)). A nested sequence of intervals therefore begins with $I_0$, and a statement about "the first $N$ terms" means the indices $0, \dots, N-1$.