Let *F* be an ordered field. We are mainly interested in the cases *F* = ℚ and *F* = ℝ though the arguments go through in greater generality for an arbitrary totally ordered set.

#### Theorem 3.1

*A sequence* 〈*u*_{n}〉 *of elements of* *F* *necessarily contains a subsequence* 〈*u*_{nk}〉 *such that either* *u*_{nk} ≥ *u*_{nℓ} *whenever* *k* > ℓ, *or* *u*_{nk} ≤ *u*_{nℓ} *whenever* *k* > ℓ.

This is an immediate consequence of the following more detailed result.

#### Theorem 3.2

*Let* *u* ∈ ^{∗}F = *F*^{ℕ}/𝓤 *be the element obtained as the equivalence class of the sequence* 〈*u*_{n}〉. *Consider the partition* ℕ = *A* ⊔ *B* ⊔ *C* *where A* = {*n* ∈ ℕ: *u*_{n} < *u*}, *B* = {*n* ∈ ℕ: *u*_{n} = *u*}, *C* = {*n* ∈ ℕ: *u*_{n} > *u*}. *Then exactly one of the following three possibilities occurs*:

*B* ∈ 𝓤 *and then* 〈*u*_{n}〉 *contains an infinite constant subsequence*;

*A* ∈ 𝓤 *and then* 〈*u*_{n}〉 *contains an infinite strictly increasing subsequence*;

*C* ∈ 𝓤 *and then* 〈*u*_{n}〉 *contains an infinite strictly decreasing subsequence*.

#### Proof

By the property of an ultrafilter, exactly one of the sets *A*, *B*, *C* is in 𝓤. If *B* ∈ 𝓤 then *u* is an element of the subfield *F* ⊆ ^{∗}F (embedded via constant sequences). Since *B* ⊆ ℕ is necessarily infinite, enumerating it we obtain the desired subsequence.

Now assume *A* ∈ 𝓤. We choose any element *u*_{n1} ∈ *A* to be the first term in the subsequence. We then inductively choose the index *n*_{k+1} > *n*_{k} in *A* so that *u*_{nk+1} is the earliest term greater than *u*_{nk} and therefore closer to *u* than the previous term *u*_{nk} If the subsequence were to terminate at, say, *u*_{p}, this would imply that {*n* ∈ ℕ: *u*_{n} ≤ *u*_{p}} ∈ 𝓤 and therefore *u* ≤ *u*_{p}, contradicting the definition of the set *A*. Therefore we necessarily obtain an infinite increasing subsequence.

The case *C* ∈ 𝓤 is similar and results in a decreasing sequence. □

The approach exploiting ^{∗}F has the advantage that the proof does not require constructing a completion of the field in the case *F* = ℚ. To work with the ultrapower, one needs neither advanced logic nor a crash course in NSA, since the ultrapower construction involves merely quotienting by a maximal ideal as is done in any serious undergraduate algebra course (see Section 2).

A monotone sequence can also be chosen by the following more traditional consideration. If the sequence is unbounded, one can choose a sequence that diverges to infinity. If the sequence is bounded, one applies the Bolzano-Weierstrass theorem (*each bounded sequence has a convergent subsequence*) to extract a convergent subsequence. Finally, a convergent sequence contains a monotone one by analyzing the terms lying on one side of the limit (whichever side has infinitely many terms).

The proof via an ultrapower allows one to bypass the issue of convergence. Once one produces a monotone subsequence, it will also be convergent in the bounded case but only when the field is complete. Furthermore, one avoids the use of the Bolzano–Weierstrass theorem.

Since in the case of *F* = ℚ the Bolzano–Weierstrass theorem is inapplicable, one would need first to complete ℚ to ℝ by an analytic procedure which is arguably at least as complex as the algebraic construction involved in the ultrapower of Section 2.

There is a clever proof of the same result, as follows (see e.g., problem 6 on page 4 in Newman [3]). Call a term in the sequence a *peak* if it is larger than everything which comes after it. If there are infinitely many peaks, they form an infinite decreasing subsequence. If there are finitely many peaks, start after the last one. From here on every term has a larger term after it, so one inductively forms an increasing subsequence (from this lemma one derives a simple proof of the Bolzano–Weierstrass theorem).

While the proof in Newman does not rely on an ultrapower, the idea of the ultrapower proof is more straightforward once one is familiar with the ultrapower construction, since it is natural to plug a sequence into it and examine the consequences.

We provide another illustration of how the element *u* = [〈*u*_{n}〉] can serve as an organizing principle that allows us to detect properties of monotone subsequences. To fix ideas let *F* = ℝ. An element *u* ∈ ^{∗}ℝ is called finite if –*r* < *u* < *r* for a suitable *r* ∈ ℝ. Let ^{𝔥}ℝ ⊆ ^{∗}ℝ be the subring of finite elements of ^{∗}ℝ. The standard part function **st**: ^{𝔥}ℝ → ℝ rounds off each finite hyperreal *u* to its nearest real number *u*_{0} = **st**(*u*).

#### Proposition 3.5

*If* *u* ∈ 𝔥ℝ *and u* > *u*_{0} *then the sequence* 〈*u*_{n}〉 *possesses a strictly decreasing subsequence*.

#### Proof

Since *u* > *u*_{0} we have {*n* ∈ ℕ: *u*_{n} > *u*_{0}} ∈ 𝓤. We start with an arbitrary *n*_{1} ∈ {*n* ∈ ℕ: *u*_{n} > *u*_{0}} and inductively choose *n*_{k+1} so that *u*_{nk+1} is closer to *u* than *u*_{nk}. We argue as in the proof of Theorem 3.2 to show that the process cannot terminate and therefore produces an infinite subsequence. □

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.