aa r X i v : . [ m a t h . C A ] M a r MONOTONE SUBSEQUENCE VIA ULTRAPOWER
PIOTR B LASZCZYK, VLADIMIR KANOVEI, MIKHAIL G. KATZ,AND TAHL NOWIK
Abstract.
An ultraproduct can be a helpful organizing principlein presenting solutions of problems at many levels, as argued byTerence Tao. We apply it here to the solution of a calculus problem:every infinite sequence has a monotone infinite subsequence, andgive other applications.Keywords: ordered structures; monotone subsequence; ultra-power; saturation; compactness Introduction
Solutions to even elementary calculus problems can be tricky butin many cases, enriching the foundational framework available enablesone to streamline arguments, yielding proofs that are more naturalthan the traditionally presented ones.We explore various proofs of the elementary fact that every infinitesequence has a monotone infinite subsequence, including some thatproceed without choosing a convergent one first.An ultraproduct can be a helpful organizing principle in presentingsolutions of problems at many levels, as argued by Terence Tao in[1]. We apply it here to the solution of the problem mentioned above.A related but different problem of proving that every infinite totallyordered set contains a monotone sequence is treated by Hirshfeld in [2,Exercise 1.2, p. 222]. We first present the ultrapower construction inSection 2. Readers familiar with ultraproducts can skip ahead to theproof in Section 3.2.
Ultrapower construction
Let us outline a construction (called an ultrapower ) of a hyperrealextension R ֒ → ∗ R exploited in our solution in Section 3. Let R N denote the ring of sequences of real numbers, with arithmetic operationsdefined termwise. Then we have a totally ordered field ∗ R = R N / MAXwhere “MAX” is a suitable maximal ideal. Elements of ∗ R are called Mathematics Subject Classification.
Primary 26A06; Secondary 26A48,26E35, 40-99. hyperreal numbers. Note the formal analogy between the quotient ∗ R = R N / MAX and the construction of the real numbers as equivalenceclasses of Cauchy sequences of rational numbers. In both cases, thesubfield is embedded in the superfield by means of constant sequences,and the ring of sequences is factored by a maximal ideal .We now describe a construction of such a maximal ideal MAX ⊆ R N exploiting a suitable finitely additive measure ξ : P ( N ) → { , } (thus ξ takes only two values, 0 and 1) taking the value 1 on each cofinite set, where P ( N ) is the set of subsets of N . The ideal MAX consists ofall “negligible” sequences h u n i , i.e., sequences which vanish for a setof indices of full measure ξ , namely, ξ (cid:0) { n ∈ N : u n = 0 } (cid:1) = 1. Thesubset U = U ξ ⊆ P ( N ) consisting of sets of full measure ξ is called afree ultrafilter (these can be shown to exist using Zorn’s lemma). Asimilar construction applied to Q produces the field ∗ Q of hyperrationalnumbers. The construction can also be applied to a general orderedset F to obtain an ultrapower extension denoted ∗ F = F N / U . Definition 2.1.
The order on the field ∗ F is defined by setting[ h u n i ] < [ h v n i ] if and only if ξ ( { n ∈ N : u n < v n } ) = 1or equivalently { n ∈ N : u n < v n } ∈ U .In particular, every element x ∈ F is canonically identified with theclass [ h x i ] of the constant sequence h x i with general term x . Then x ∈ ∗ F satisfies x < v if and only if { n ∈ N : x < v n } ∈ U .3. Solution
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 generalityfor an arbitrary totally ordered set. Theorem 3.1.
A sequence h u n i of elements of F necessarily contains asubsequence h u n k i such that either u n k ≥ u n ℓ whenever k > ℓ , or u n k ≤ u n ℓ whenever k > ℓ . This is an immediate consequence of the following more detailedresult.
Theorem 3.2.
Let u ∈ ∗ F = F N / U be the element obtained as theequivalence class of the sequence h u n i . Consider the partition N = A ⊔ B ⊔ C where A = { n ∈ N : u n < u } , B = { n ∈ N : u n = u } , C = For each pair of complementary infinite subsets of N , such a measure ξ “decides” ina coherent way which one is “negligible” (i.e., of measure 0) and which is “dominant”(measure 1). ONOTONE SUBSEQUENCE VIA ULTRAPOWER 3 { n ∈ N : u n > u } . Then exactly one of the following three possibilitiesoccurs: (1) B ∈ U and then h u n i contains an infinite constant subsequence; (2) A ∈ U and then h u n i contains an infinite strictly increasingsubsequence; (3) C ∈ U and then h u n i contains an infinite strictly decreasingsubsequence.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 (em-bedded 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 n ∈ A to be the firstterm in the subsequence. We then inductively choose the index n k +1 >n k in A so that u n k +1 is the earliest term greater than u n k and thereforecloser to u than the previous term u n k . If the subsequence were toterminate at, say, u p , this would imply that { n ∈ N : u n ≤ u p } ∈ U andtherefore u ≤ u p , contradicting the definition of the set A . Thereforewe necessarily obtain an infinite increasing subsequence.The case C ∈ U is similar and results in a decreasing sequence. (cid:3) Remark 3.3.
The proof is essentially a two-step procedure: (1) weplug the sequence into the ultrapower construction, producing an ele-ment u ∈ ∗ F ; (2) in each of the cases specified by the element u , weinductively find a monotone subsequence.The approach exploiting ∗ F has the advantage that the proof doesnot require constructing a completion of the field in the case F = Q .To work with the ultrapower, one needs neither advanced logic nor acrash course in NSA, since the ultrapower construction involves merelyquotienting by a maximal ideal as is done in any serious undergraduatealgebra course (see Section 2).A monotone sequence can also be chosen by the following more tra-ditional consideration. If the sequence is unbounded, one can choosea sequence that diverges to infinity. If the sequence is bounded, oneapplies the Bolzano-Weierstrass theorem ( each bounded sequence hasa convergent subsequence ) to extract a convergent subsequence. Fi-nally, a convergent sequence contains a monotone one by analyzing theterms lying on one side of the limit (whichever side has infinitely manyterms).The proof via an ultrapower allows one to bypass the issue of con-vergence. Once one produces a monotone subsequence, it will also be P. B LASZCZYK, V. KANOVEI, M. KATZ, AND T. NOWIK 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 in-applicable, one would need first to complete Q to R by an analyticprocedure which is arguably at least as complex as the algebraic con-struction involved in the ultrapower of Section 2.There is a clever proof of the same result, as follows (see e.g., prob-lem 6 on page 4 in Newman [3]). Call a term in the sequence a peak ifit is larger than everything which comes after it. If there are infinitelymany peaks, they form an infinite decreasing subsequence. If thereare finitely many peaks, start after the last one. From here on everyterm has a larger term after it, so one inductively forms an increas-ing subsequence (from this lemma one derives a simple proof of theBolzano–Weierstrass theorem). Remark 3.4.
The proof in Newman consists of two steps: (1) in-troduce the idea of a peak; (2) consider separately the cases whenthe number of peaks is finite or infinite to produce the desired mono-tone subsequence. While the basic structure of the proof is similarto that using the ultrapower (see Remark 3.3), the basic difference isthat step (1) in Newman is essentially ad-hoc, is tailor-made for thisparticular problem, and is not applicable to solving other problems.Meanwhile the ultrapower construction is applicable in many other sit-uations (see e.g., Section 4).While the proof in Newman does not rely on an ultrapower, the ideaof the ultrapower proof is more straightforward once one is familiarwith the ultrapower construction, since it is natural to plug a sequenceinto it and examine the consequences.We provide another illustration of how the element u = [ h u n i ] canserve as an organizing principle that allows us to detect properties ofmonotone subsequences. To fix ideas let F = R . An element u ∈ ∗ R iscalled finite if − r < u < r for a suitable r ∈ R . Let h R ⊆ ∗ R be thesubring 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 = st ( u ). Proposition 3.5. If u ∈ h R and u > u then the sequence h u n i pos-sesses a strictly decreasing subsequence.Proof. Since u > u we have { n ∈ N : u n > u } ∈ U . We start with anarbitrary n ∈ { n ∈ N : u n > u } and inductively choose n k +1 so that u n k +1 is closer to u than u n k . We argue as in the proof of Theorem 3.2to show that the process cannot terminate and therefore produces aninfinite subsequence. (cid:3) ONOTONE SUBSEQUENCE VIA ULTRAPOWER 5 Compactness
A more advanced application is a proof of the nested decreasing se-quence property for compact sets (Cantor’s intersection theorem) usingthe property of saturation . Such a proof exbibits compactness as closelyrelated to the more general property of saturation, shedding new lighton the classic property of compactness.A typical proof of
Cantor’s intersection theorem for a nested de-creasing sequence of compact subsets A n ⊆ R would use the monotonesequence h u n i where u n is the minimum of each A n . We will present adifferent and more conceptual proof.Each set A ⊆ R has a natural extension denoted ∗ A ⊆ ∗ R . Similarlythe powerset P = P ( R ) has a natural extension ∗ P identified with aproper subset of P ( ∗ R ). Each element of ∗ P is naturally identified witha subset of ∗ R called an internal set .The principle of saturation holds for arbitrary nested decreasing se-quences of internal sets but we will present it in a following specialcase. Theorem 4.1 (Saturation) . If h A n : n ∈ N i is a nested decreasingsequence of nonempty subsets of R then the sequence h ∗ A n : n ∈ N i hasa common point.Proof. Let P = P ( R ) be the set of subsets of R . We view the se-quence h A n ∈ P : n ∈ N i as a function f : N → P , n A n . By theextension principle we have a function ∗ f : ∗ N → ∗ P . Let B n = ∗ f ( n ).For each finite n we have B n = ∗ A n ∈ ∗ P . For each infinite value of theindex n = H the entity B H ∈ ∗ P is by definition internal but is not(necessarily) the natural extension of any subset of R .If h A n i is a nested sequence in P then by transfer h B n : n ∈ ∗ N i is anested sequence in ∗ P with each B n nonempty. Let H be a fixed infiniteindex. Then for each finite n the set ∗ A n ⊆ ∗ R includes B H . Chooseany element c ∈ B H . Then c is contained in ∗ A n for each finite n sothat c ∈ T n ∈ N ∗ A n as required. (cid:3) Remark 4.2.
An equivalent formulation of Theorem 4.1 is as follows.If the family of subsets { A n } n ∈ N has the finite intersection propertythen ∃ c ∈ T n ∈ N ∗ A n .Let X be a topological space. Let p ∈ X . The halo of p , denoted h ( p )is the intersection of all ∗ U where U runs over all neighborhoods of p in X (a neighborhood of p is an open set that contains p ). A point y ∈ ∗ X is called nearstandard in X if there is p ∈ X such that y ∈ h ( p ). P. B LASZCZYK, V. KANOVEI, M. KATZ, AND T. NOWIK
Theorem 4.3.
A space X is compact if and only if every y ∈ ∗ X isnearstandard in X .Proof. To prove the direction ⇒ , assume X is compact, and let y ∈ ∗ X .Let us show that y is nearstandard (this direction does not requiresaturation). Assume on the contrary that y is not nearstandard. Thismeans that it is not in the halo of any point p ∈ X . This meansthat every p ∈ X has a neighborhood U p such that y ∗ U p . Thecollection { U p } p ∈ X is an open cover of X . Since X is compact, thecollection has a finite subcover U p , . . . , U p n , so that X = U p ∪ . . . ∪ U p n . But for a finite union, the star of union is the union of stars.Thus ∗ X is the union of ∗ U p , . . . , ∗ U p n , and so the point y is in one ofthe sets ∗ U p , . . . , ∗ U p n , a contradiction.Next we prove the direction ⇐ (this direction exploits saturation).Assume every y ∈ ∗ X is nearstandard, and let { U a } be an open coverof X . We need to find a finite subcover.Assume on the contrary that the union of any finite collection of U a is not all of X . Then the complements of U a are a collection of (closed)sets { S a } with the finite intersection property. It follows that the collec-tion { ∗ S a } similarly has the finite intersection property. By saturation(see Remark 4.2), the intersection of all ∗ S a is non-empty. Let y be apoint in this intersection. Let p ∈ X be such that y ∈ h ( p ). Now { U a } is a cover of X so there is a U b such that p ∈ U b . But y is in ∗ S a for all a ,in particular y ∈ ∗ S b , so it is not in ∗ U b , a contradiction to y ∈ h ( p ). (cid:3) Theorem 4.4 (Cantor’s intersection theorem) . A nested decreasingsequence of nonempty compact sets has a common point.Proof.
Given a nested sequence of compact sets K n , we consider thecorresponding decreasing nested sequence of internal sets, h ∗ K n : n ∈ N i . This sequence has a common point x by saturation. But for acompact set K n , every point of ∗ K n is nearstandard (i.e., infinitelyclose to a point of K n ) by Theorem 4.3. In particular, st ( x ) ∈ K n forall n , as required. (cid:3) More advanced applications can be found in [4, 5, 6].
References [1] Tao T., Hilbert’s fifth problem and related topics, Graduate Studies in Math-ematics, 153, American Mathematical Society, Providence, RI, 2014.[2] Hirshfeld J., Nonstandard combinatorics, Studia Logica, 1988, 47, no. 3, 221–232.[3] Newman D., A problem seminar, Problem Books in Mathematics, Springer-Verlag, New York-Berlin, 1982.
ONOTONE SUBSEQUENCE VIA ULTRAPOWER 7 [4] Fletcher P., Hrbacek K., Kanovei V., Katz M., Lobry C., SandersS., Approaches to analysis with infinitesimals following Robinson, Nel-son, and others, Real Analysis Exchange 2017, 42, no. 2, 193–252. See https://arxiv.org/abs/1703.00425 and http://msupress.org/journals/issue/?id=50-21D-61F [5] Herzberg F., Kanovei V., Katz M., Lyubetsky V., Minimal axiomatic frame-works for definable hyperreals with transfer, Journal of Symbolic Logic,to appear. See https://arxiv.org/abs/1707.00202 and http://dx.doi.org/10.1017/jsl.2017.48 [6] Nowik T., Katz M., Differential geometry via infinitesimal displacements,Journal of Logic and Analysis, 2015, 7, no. 5, 1–44. See and http://arxiv.org/abs/1405.0984
P. B laszczyk, Institute of Mathematics, Pedagogical University ofCracow, Poland
E-mail address : [email protected] V. Kanovei, IPPI, Moscow, and MIIT, Moscow, Russia
E-mail address : [email protected] M. Katz, Department of Mathematics, Bar Ilan University, RamatGan 52900 Israel
E-mail address : [email protected] T. Nowik, Department of Mathematics, Bar Ilan University, RamatGan 52900 Israel
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