Research Article Completeness of Ordered Fields and a Trio ...downloads.hindawi.com › journals › aaa › 2016 › 6023273.pdf · Research Article Completeness of Ordered Fields

Research ArticleCompleteness of Ordered Fields and a Trio ofClassical Series Tests

Robert Kantrowitz1 and Michael M Neumann2

1Department of Mathematics Hamilton College 198 College Hill Road Clinton NY 13323 USA2Department of Mathematics and Statistics Mississippi State University Mississippi State MS 39762 USA

Correspondence should be addressed to Robert Kantrowitz rkantrowhamiltonedu

Received 12 April 2016 Revised 25 September 2016 Accepted 12 October 2016

Academic Editor Bashir Ahmad

Copyright copy 2016 R Kantrowitz and M M Neumann This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This article explores the fate of the infinite series tests of Dirichlet Dedekind and Abel in the context of an arbitrary ordered field Itis shown that each of these three tests characterizes the Dedekind completeness of an Archimedean ordered field specifically noneof the three is valid in any proper subfield ofR The argument hinges on a contractive-type property for sequences in Archimedeanordered fields that are bounded and strictly increasing For an arbitrary ordered field it turns out that each of the tests of Dirichletand Dedekind is equivalent to the sequential completeness of the field

1 Introduction

The main theorems of calculus derive their validity from thecompleteness property of the real numbers but the extentof the interconnectedness of the theorems themselves iscontinuing to come into sharper focus The ldquoreal analysis inreverserdquo program as described in [1] is shedding light on thefact that so many of the theorems of real analysis are actuallyequivalent reformulations of the notion of completenessand hence of one another In addition to [1] the flurry ofrecent articles [2ndash9] is dedicated to the study of abstractordered fields particularly the various ways in which theArchimedean and completeness properties are manifested

An ordered field F has the Archimedean property if thecanonically embedded copy of the natural numbers is notbounded above Equivalent expressions of the Archimedeanproperty aremany and several of the articles cited above treatthem Discussions of completeness center around Dedekindcompleteness and Cauchy completeness The former meansthat F has the supremum property namely every nonemptysubset of F that is bounded above has a least upper boundin F the latter also called sequential completeness entails theconvergence of all Cauchy sequences in F

The ordered field F is topologized by the basic openintervals (119886 119887) = 119909 isin F 119886 lt 119909 lt 119887 for all elements 119886 119887 isin F

while the absolute value function for F is defined by |119886| =maxminus119886 119886 for all 119886 isin F Absolute value works in conjunctionwith the order topology to lend a familiarmeaning to analyticaspects of F For example a sequence (119909119899) of elements of Fconverges to 119909 isin F if for every positive element 120576 in F thereis a natural number119873 for which |119909119899minus119909| lt 120576whenever 119899 ge 119873The definition of a Cauchy sequence in F also parallels itscounterpart from real analysis

It is easy to prove that a field that is Dedekind completehas the Archimedean property and it is well known thata field that is Dedekind complete is also Cauchy completeMoreover if an Archimedean ordered field is Cauchy com-plete then it must be Dedekind complete (CA35 of [3] orSections 1-2 of [8]) such a field is up to isomorphism thefieldR of real numbers Behind all this of course is Cantorrsquosclassical construction of the real numbers as the Cauchycompletion of the rationals ([10]) A remarkable fact aboutordered fields with the Archimedean property is that theyare precisely those that are isomorphic to subfields of R(Theorem 35 of [5] or Section 4 of [1])

Series tests are among the many theorems from calculusthat have been considered in the classification of orderedfields An elementary example from the Classroom Capsule[6] exposes the equivalence of the geometric series test and

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2016 Article ID 6023273 6 pageshttpdxdoiorg10115520166023273

2 Abstract and Applied Analysis

the Archimedean property Other familiar tests that enter thefray are given as follows

Absolute Convergence Test If (119886119899) is a sequence of elements inthe ordered field F for which the seriessum |119886119899| converges thenthe series sum119886119899 also convergesComparisonTest If (119886119899) and (119887119899) are sequences of nonnegativeelements in the ordered field F that satisfy 119886119899 le 119887119899 for all 119899 andsum119887119899 converges then sum119886119899 converges

We document the connection between completeness andthese two series tests in the following theorem which maybe viewed as a modest complement to the main result of [2]Theorem 1 will be an important tool in this article

Theorem1 For an ordered field F the following statements areequivalent

(a) F is Cauchy complete(b) The absolute convergence test holds(c) The comparison test holds

Proof The fact that the absolute convergence test character-izes Cauchy completeness for ordered fields is proven in thearticle [2]

The validity of the comparison test in fields that areCauchy complete follows by noting that convergence of theseries sum119887119899 implies that its sequence (119861119899) of partial sums isconvergent and hence Cauchy The sequence (119860119899) of partialsums of the series sum119886119899 is therefore also Cauchy from whichthe convergence of the series sum119886119899 follows from the Cauchycompleteness of the field

Finally to see that the absolute convergence test holdswhenever the comparison test does observe that for all 119899the inequalities

0 le 119886119899 + 10038161003816100381610038161198861198991003816100381610038161003816 le 2 10038161003816100381610038161198861198991003816100381610038161003816 (1)

are true in F setting the stage for the comparison test toensure that the seriessum(119886119899+|119886119899|) converges Since alsosumminus|119886119899|converges it follows that sum119886119899 converges

Theorem 1 refines a result of [8] on the characteriza-tion of Dedekind complete ordered fields in terms of theArchimedean property together with either (b) or (c)

The equivalence of statements (a) and (b) of Theorem 1appears mutatis mutandis in functional analysis where itis known that a normed linear space is a Banach space ifand only if the absolute convergence test for series is valid(Theorem 28 of [11] or Statement (VIII) of [12])

Our goal in the present article is to involve the series testsof Dirichlet Dedekind and Abel in the taxonomy of orderedfieldsThe classical versions of these may be found for exam-ple in [13] or [14] but each has meaning in the more generalabstract context We devote Section 2 to their statements anda short discussion of them Like the absolute convergencetest and the comparison test Dirichletrsquos Dedekindrsquos andAbelrsquos tests are connected with completeness in the sense ofboth Dedekind and Cauchy These relationships are detailed

in Sections 4 and 5 Along the way we explore the role ofsequences of bounded variation in ordered fields

2 The Series Tests of DirichletDedekind and Abel

The hypothesis of Dedekindrsquos test calls for the sequence (119887119899)to have bounded variationwhich in the setting of the orderedfield F means that there is an element 119872 isin F for which theinequality sum119899119896=1 |119887119896+1 minus 119887119896| le 119872 holds for all 119899 isin N We willtreat sequences of bounded variation in some detail in thenext section Without further ado here is the trio of seriestests under consideration for an arbitrary ordered field

Dirichletrsquos Test If sum119886119899 is a series whose partial sums forma bounded sequence and (119887119899) is a decreasing sequence thatconverges to 0 then the seriessum119886119899119887119899 convergesDedekindrsquos Test If sum119886119899 is a convergent series and (119887119899) is asequence of bounded variation then the seriessum119886119899119887119899 conver-ges

Abelrsquos Test Ifsum119886119899 is a convergent series and (119887119899) is amonotoneconvergent sequence then the series sum119886119899119887119899 converges

Dirichletrsquos test is recognizable as a generalization ofLeibnizrsquos alternating series test from calculus (Theorem 342of [14]) and as Apostol mentions in Section 815 of [13] thesethree tests are useful tools when one is confronted with thetask of trying to determine convergence of real series thatdo not converge absolutely Moreover as Knopp points outin Section 55 of [14] a common feature of the tests in thistrio is that they all allow conclusions to be drawn about theseries sum119886119899119887119899 from assumptions concerning the series sum119886119899and the sequence (119887119899) Knopp remarks that such tests may beconstrued as ldquocomparison tests in the extended senserdquo andhe also includes a few applications and examples of them

It turns out that Abelrsquos test can be inferred fromDedekindrsquos test since as explained in the next section everymonotone convergent sequence has bounded variation Amomentrsquos reflection reveals that Abelrsquos test is also a corollaryof Dirichletrsquos test Indeed the sequence of partial sums of aconvergent series sum119886119899 is bounded so attention needs onlyto be directed to the stipulation that (119887119899) is a monotoneconvergent sequence If (119887119899) is decreasing and its limit is 119887then the auxiliary sequence given by 119888119899 = 119887119899 minus 119887 is alsodecreasing and it converges to 0 Dirichletrsquos test thus ensuresthat the seriessum119886119899119888119899 convergesThe convergence of the seriessum119886119899119887119899 follows easily from algebraic properties of convergentseries The case that (119887119899) is increasing with limit 119887 is handledanalogously since this time the sequence (minus119888119899) decreases to 0

None of the series tests of Dedekind Dirichlet or Abel ison its own strong enough to imply Dedekind completenessfor an ordered field F in which it holds If F is Archimedeanhowever and any of these three tests is in force then it willfollow from Theorem 3 that F is Dedekind complete Thelinchpin is a contractive-type property for certain sequencesin Archimedean ordered fields which is detailed in Lemma 4

Abstract and Applied Analysis 3

Aswe shall see inTheorem6 in the final section the valid-ity of either of the tests of Dedekind orDirichlet in an orderedfield is equivalent to the Cauchy completeness of the field Ina similar vein it was recently confirmed in [15] that a normedlinear space is a Banach space precisely when a suitable vectorversion of Dedekindrsquos test holds and moreover that a unitalnormed algebra is a Banach algebra if and only if an algebraversion of Dedekindrsquos test is valid

3 Sequences of Bounded Variation

The set of all sequences of elements of the ordered field F

having bounded variation will be denoted by 119887VF Examplesabound Indeed sequences that are monotone and boundedautomatically have bounded variation if (119886119899)119899isinN is increasingfor example with |119886119899| le 119870 for all 119899 isin N then

119899

sum119896=1

1003816100381610038161003816119886119896+1 minus 1198861198961003816100381610038161003816 =119899

sum119896=1

(119886119896+1 minus 119886119896) = 119886119899+1 minus 1198861

le 1003816100381610038161003816119886119899+11003816100381610038161003816 minus 1198861 le 119870 minus 1198861(2)

which shows that (119886119899)119899isinN has bounded variationSequences of bounded variation are always bounded

since for any 119899 isin N10038161003816100381610038161198861198991003816100381610038161003816 le 100381610038161003816100381611988611003816100381610038161003816 + 10038161003816100381610038161198862 minus 11988611003816100381610038161003816 + 10038161003816100381610038161198863 minus 11988621003816100381610038161003816 + sdot sdot sdot + 1003816100381610038161003816119886119899 minus 119886119899minus11003816100381610038161003816

le 100381610038161003816100381611988611003816100381610038161003816 + 119872(3)

But in general there is no relationship between 119887VF and theset 119888F of convergent sequences in F For example in anyArchimedean ordered field the sequence

1 0 12 013 0

14 0 (4)

converges to 0 but it does not have bounded variationwhereas in any non-Archimedean ordered field it hasbounded variation but does not converge In fact in a non-Archimedean ordered field any sequence of rational numbershas bounded variation

Like their counterparts in the realm of functions on aninterval (see Section 67 of [13]) sequences of bounded vari-ation admit a Jordan decomposition Specifically a sequence(119886119899)119899isinN in 119887VF may be resolved into the difference of twobounded increasing sequences (119901119899)119899isinN and (119902119899)119899isinN in acanonical way Indeed it is easily verified that this is accom-plished by the choice 119901119899 = (V119899+119886119899)2 and 119902119899 = (V119899minus119886119899)2 forall 119899 isin N where V1 = 0 and V119899 = sum119899minus1119896=1 |119886119896+1 minus 119886119896| for all 119899 ge 2

Since by Proposition 4 of [9] the Archimedean propertyof an ordered field F is characterized by the condition thatall increasing sequences that are bounded above are Cauchysequences the Jordan decomposition now reveals that F hasthe Archimedean property if and only if every sequence ofbounded variation in F is a Cauchy sequence

Moreover if every sequence of bounded variation inF happens to converge then because we have seen thatevery monotone bounded sequence lies in 119887VF themonotone

convergence theorem holds for F The monotone convergencetheorem is a stalwart among the statements that are equiv-alent to Dedekind completeness (CA13 of [3] or Section 1of [8]) so we conclude that a field for which the inclusion119887VF sube 119888F holds must be Dedekind complete

The hypotheses for Abelrsquos test seem to vary slightly in theliterature The version we work with follows Theorem 829of [13] in requiring a convergent series sum119886119899 and a monotoneconvergent sequence (119887119899) However in the statement of Abelrsquostest that is Theorem 551 of [14] the series sum119886119899 is stillrequired to be convergent whereas the sequence (119887119899) is onlyassumed to be monotone and bounded The two versions areequivalent preciselywhen the underlying field isRMoreoveron account of the Jordan decomposition it turns out that inany field the latter version is equivalent to Dedekindrsquos test

As we see facts about sequences of bounded variationmay often be deduced from properties of monotone sequen-ces Though they find themselves somewhat overshadowedbymonotone sequences sequences of bounded variation stillplay a role in analysis It turns out that Dedekindrsquos test is theeasier half of the result that says that the topological dualspace of the Banach space 119888119904R of all real sequences (119886119899) forwhich the associated series sum119886119899 converges may be identifiedwith 119887VR (Exercise IV1312 of [16]) Sequences of boundedvariation rear their heads again in the related context ofsummability theory in order that an infinite matrix definesa transformation that maps 119888119904R into 119888119904R one of the necessaryand sufficient conditions is that each of its rows is in 119887VR(Theorem 561 of [14] or Exercise II445 of [16])Lemma 2 If the ordered field F is Cauchy complete then theseries tests of Dirichlet Dedekind and Abel all hold

Proof The identity for summation by parts due to Abel(Theorem 827 of [13]) a well-known analogue for integrationby parts in a Riemann-Stieltjes integral carries over to thesetting of an abstract field (or in fact of any ring) verbatimfor elements 1198861 119886119899 and 1198871 119887119899 of F and 119860119896 = sum119896119895=1 119886119895for 119896 = 1 119899 the following equality holds

119899

sum119896=1

119886119896119887119896 = 119860119899119887119899+1 minus119899

sum119896=1

119860119896 (119887119896+1 minus 119887119896) (5)

As in the classical approach it seems natural to deducethe convergence of the series sum119886119899119887119899 by establishing theconvergence of the two sequences on the right hand side ofthe preceding formula

In the case of Dirichletrsquos test this method turns out tobe successful in an arbitrary Cauchy complete field thankstoTheorem 1 Indeed the hypotheses of Dirichletrsquos test easilyensure that (119860119899119887119899+1) and sum |119887119896+1 minus 119887119896| both converge Henceif 119872 denotes a positive upper bound on the sequence ofpartial sums (119860119899) then the seriessuminfin119896=1119872|119887119896+1minus119887119896| convergesBy Theorem 1 we obtain the convergence of the seriessuminfin119896=1 |119860119896(119887119896+1 minus 119887119896)| from the comparison test and then thedesired convergence of the seriessuminfin119896=1 119860119896(119887119896+1 minus 119887119896) from theabsolute convergence test


However more work is needed for Dedekindrsquos test sincethe convergence of sequences of bounded variation is guaran-teed only in the setting of a Dedekind complete ordered fieldWe therefore separate the following two cases

If the Cauchy complete field F happens to be Archime-dean then F is Dedekind complete and we are in the classicalsetting F = R for which Dedekindrsquos and Abelrsquos tests areof course immediate from the preceding partial summationformula

It thus remains to address the case that F is non-Archi-medean Since the hypotheses for Dedekindrsquos and Abelrsquos testsensure that 119886119899 rarr 0 as 119899 rarr infin and that the sequence (119887119899) isbounded we have that 119886119899119887119899 rarr 0 as 119899 rarr infin Convergenceof the series sum119886119899119887119899 thus follows from part (b) of the maintheorem of [2]

4 Dedekind Completeness and Series Tests

We now involve the trio of series tests in a list of statementsabout an ordered field that are equivalent to Dedekindcompleteness

Theorem 3 For an ordered field F the following statementsare equivalent

(a) F is Dedekind complete(b) Sequences in F that are monotone and bounded are

convergent(c) 119887VF sube 119888F (d) F is Archimedean and Dirichletrsquos test holds(e) F is Archimedean and Dedekindrsquos test holds(f) F is Archimedean and Abelrsquos test holds

As indicated in the previous section the equivalence ofstatements (a) and (b) is well known and the equivalenceof (b) and (c) is clear from the Jordan decomposition forsequences of bounded variation

Because a field that is Dedekind complete is Archimedeanand Cauchy complete the implications (a) rArr (d) and (a) rArr(e) follow from Lemma 2 The implications (d) rArr (f) and(e) rArr (f) are consequences of the fact that Dirichletrsquos test andDedekindrsquos test both imply Abelrsquos test as detailed toward theend of Section 2

It thus remains to establish that (f) rArr (b) Our proof willbe based on the following result

Lemma 4 Let 119903 be a positive element of the Archimedeanordered field F and let (119888119899)119899isinN be a strictly increasing boundedsequence in F Then there exists a subsequence (119888119899119896)119896isinN suchthat

119888119899119896+2 minus 119888119899119896+1 lt 119903 (119888119899119896+1 minus 119888119899119896) forall119896 isin N (6)

Proof By the characterization of Archimedean fields men-tioned earlier we may and do assume that F is a subfield ofR Then there exists an element 119871 isin R for which 119888119899 rarr 119871 as119899 rarr infin The inductive choice of the desired subsequence will

be straightforward once we know that for every 119901 isin N thereexists some 119902 isin N with 119902 gt 119901 such that the estimate

119888119895 minus 119888119902 lt 119903 (119888119902 minus 119888119901) (7)

holds for all 119895 isin Nwith 119895 gt 119902 For this fix119901 isin N and note thatwhenever 119902 isin N satisfies 119902 gt 119901 the inequality 119903(119888119901+1 minus 119888119901) le119903(119888119902 minus 119888119901) automatically holds Now take 119902 isin N so large that119902 gt 119901 and 119871 minus 119888119902 lt 119903(119888119901+1 minus 119888119901)Then for all 119895 isin N with 119895 gt 119902we obtain

119888119895 minus 119888119902 lt 119871 minus 119888119902 lt 119903 (119888119901+1 minus 119888119901) le 119903 (119888119902 minus 119888119901) (8)

To complete the proof of the lemma we choose 1198991 = 1and then by induction a sequence of integers 119899119896 isin N with119899119896 lt 119899119896+1 such that the a priori estimate

119888119895 minus 119888119899119896+1 lt 119903 (119888119899119896+1 minus 119888119899119896) (9)

holds for all 119896 isin N and 119895 isin N with 119895 gt 119899119896+1 Once asubsequence (119888119899119896)119896isinN with the preceding property has beenchosen for each 119896 isin N we may take 119895 = 119899119896+2 to conclude thatthe sequence satisfies the conclusion of the lemma

We mention in passing that the preceding result remainsvalid for increasing rather than strictly increasing sequencesprovided that the strict inequality lt in the assertion isreplaced byle Indeed this is obvious in the case of an increas-ing sequence that is eventually constant while every increas-ing bounded sequence that fails to be eventually constantadmits a strictly increasing subsequence to which Lemma 4may be applied

We are now ready to finish the proof of Theorem 3

Proof of (119891) rArr (119887) Suppose that F is an Archimedeanordered field for which Abelrsquos test holds and consider anarbitrary increasing bounded sequence (119888119899)119899isinN in F To seethat this sequence converges in F we may suppose that it isnot eventually constant and thus admits a strictly increasingsubsequence Since an increasing sequence converges in F

precisely when it contains a convergent subsequence wemaywithout loss of generality assume that (119888119899)119899isinN is actuallystrictly increasing We then apply Lemma 4 with the choice119903 = 14 to obtain a subsequence (119888119899119896)119896isinN for which

119888119899119896+2 minus 119888119899119896+1 lt14 (119888119899119896+1 minus 119888119899119896) (10)

for all 119896 isin N As before it suffices to show that thissubsequence converges in F This will now be established byan application of Abelrsquos test and the Archimedean propertyfor the field F For this we introduce

119886119896 = 12119896

119887119896 = 2119896 (119888119899119896+1 minus 119888119899119896)(11)

for all 119896 isin N Because F is Archimedean we know from [6]that the geometric series

infin

sum119896=1

119886119896 =infin

sum119896=1

(12)119896

(12)


converges in F Moreover the choice of the subsequence(119888119899119896)119896isinN ensures that

119887119896+1 = 2119896+1 (119888119899119896+2 minus 119888119899119896+1) lt 2119896minus1 (119888119899119896+1 minus 119888119899119896) = 12119887119896 (13)

and therefore 0 lt 119887119896+1 lt (12119896)1198871 for all 119896 isin N Thus(119887119896) decreases to 0 By Abelrsquos test we now conclude thatthe sequence of partial sums sum119898119896=1 119886119896119887119896 converges in F Sincetelescoping shows that

119898

sum119896=1

119886119896119887119896 =119898

sum119896=1

(119888119899119896+1 minus 119888119899119896) = 119888119899119898+1 minus 1198881198991 (14)

for all 119898 isin N this confirms the desired convergence of(119888119899119896)119896isinN

While the geometric series test is certainly not on its ownstrong enough to guarantee completeness the ClassroomCapsule [6] together with Theorem 3 shows that if F is anordered field in which the geometric series test along withany of the tests of Dirichlet Dedekind or Abel holds then F

is Dedekind complete Theorem 3 also reveals that the onlysubfield of R in which Dirichletrsquos Dedekindrsquos or Abelrsquos testholds is the fieldR of real numbers itself There are howevermany non-Archimedean ordered fields in which each of thesethree tests holds We will take up this issue after an example

Example 5 An example that simultaneously illustrates thefailure of each of the three tests in the field Q of rationalnumbers is provided by the geometric series suminfin119899=1 12119899 andthe harmonic sequence (1119899)119899isinN The series converges inQ so its partial sums are bounded and the sequence is adecreasing sequence that converges to 0 The hypotheses ofDirichletrsquos test are thus satisfied We invoke some calculus toshow that the seriessuminfin119899=1 1(1198992119899) does not however convergeinQThe Taylor series for (1minus119909)minus1 valid whenever |119909| lt 1 isthe familiar geometric series 1+119909+1199092+1199093+sdot sdot sdot Term-by-termintegration yields the series

119909 + 11990922 + 1199093

3 + 11990944 + sdot sdot sdot = minus log (1 minus 119909) (15)

where the equality is valid for the same radius of convergenceThe choice 119909 = 12 reveals suminfin119899=1 1(1198992119899) = log 2 Butlog 2 is irrational since otherwise the equation log 2 = 119901119902with 119901 119902 isin N would entail that the number 119890 is a rootof the polynomial 119909119901 minus 2119902 contradicting the fact that 119890 istranscendental a celebrated result of Hermite dating backto 1873 (Chapter 15 of [17]) The conclusion of Dirichletrsquostest thus does not hold for Q These data also show thatDedekindrsquos and Abelrsquos tests fail inQ

5 Cauchy Completeness and Series Tests

Weconcludewith a return to the case ofCauchy completenessand the following extension of Theorem 1


(a) F is Cauchy complete(b) The absolute convergence test holds(c) The comparison test holds(d) Dedekindrsquos test holds(e) Dirichletrsquos test holds

Proof The equivalence of statements (a) (b) and (c) washandled in Theorem 1 and the implications (a) rArr (d) and(a) rArr (e) were addressed in Lemma 2 By Theorem 3 itthus suffices to establish (d) rArr (b) and (e) rArr (b) in thenon-Archimedean case To this end we consider an arbitrarysequence (119909119899) in a non-Archimedean field F for which theseries sum |119909119899| converges

If (d) holds then we choose 119887119899 isin minus1 1 such that 119909119899 =119887119899|119909119899| for all 119899 isin N and observe that the sequence of partialsums sum119899119896=1 |119887119896+1 minus 119887119896| is bounded above in F since F fails tobe Archimedean Thus Dedekindrsquos test ensures that sum119909119899 =sum119887119899|119909119899| converges in F which confirms that (d) implies (b)

To prove the convergence of sum119909119899 under condition (e)we first note that the convergence of the series sum |119909119899| entailsthat the sequence of partial sums of the series sum119909119899 is aCauchy sequence So to meet our goal it suffices to showthat the sequence of partial sums of the series sum119909119899 has aconvergent subsequence This will be accomplished by thefollowing construction

The supposition thatsum |119909119899| converges implies that |119909119899| rarr0 as 119899 rarr infin and without loss of generality we assumethat the sequence (|119909119899|) is not eventually 0 Invoking theprocedure detailed in the proof of the classical monotoneconvergence theorem (Theorem 347 of [18]) we inductivelyextract a decreasing ldquopeakrdquo subsequence (|119909119899119896 |) as followsWestart by selecting 1198991 as the least positive integer 120581 for which|119909120581| ge |119909119895| for all 119895 isin N Thereafter once 119899119896 is designated119899119896+1 is chosen to be the least positive integer 120581 beyond 119899119896 forwhich |119909120581| ge |119909119895| for all 119895 gt 119899119896 Evidently |119909119899119896 | gt 0 for all119896 isin N and (|119909119899119896 |) decreases to 0

Now for each 119896 isin N let

119886119896 =119909119899119896 + sdot sdot sdot + 119909119899119896+1minus110038161003816100381610038161003816119909119899119896

10038161003816100381610038161003816(16)

and observe that

10038161003816100381610038161198861198961003816100381610038161003816 le10038161003816100381610038161003816119909119899119896

10038161003816100381610038161003816 + sdot sdot sdot + 10038161003816100381610038161003816119909119899119896+1minus11003816100381610038161003816100381610038161003816100381610038161003816119909119899119896

10038161003816100381610038161003816le 119899119896+1 minus 119899119896 (17)

The sequence of partial sums of the series sum119886119896 is thusbounded in F on account of the field F being non-Archimedean Hence by Dirichletrsquos test the series sum119886119896|119909119899119896 |converges Since

119899119898+1minus1

sum119896=1

119909119896 = 1199091 + sdot sdot sdot + 1199091198991minus1 +119898

sum119896=1

119886119896 1003816100381610038161003816100381611990911989911989610038161003816100381610038161003816 (18)

for all 119898 isin N we conclude that the sequence of partial sumsof the seriessum119909119899 has indeed a convergent subsequence so wehave reached our stated goal


An anonymous referee graciously offered the equivalenceof statements (a) (d) and (e) of Theorem 6 together with anoutline of a proof The suggested argument to establish thefact that Cauchy completeness is a consequence of Dirichletrsquostest was based on the main theorem of [2] along with resultsabout rearrangements of series with nonnegative terms Inthe end we chose our own version of the proof of (e) rArr (b)on account of its direct focus on the absolute convergencetest The question of whether the validity of Abelrsquos test inan arbitrary ordered field implies that the field is Cauchycomplete remains open

Competing Interests

The authors declare that they have no competing interests

References

[1] J Propp ldquoReal analysis in reverserdquo American MathematicalMonthly vol 120 no 5 pp 392ndash408 2013

[2] P L Clark and N J Diepeveen ldquoAbsolute convergence inordered fieldsrdquoAmericanMathematicalMonthly vol 121 no 10pp 909ndash916 2014

[3] M Deveau and H Teismann ldquo72 + 42 characterizations of thecompleteness and Archimedean properties of ordered fieldsrdquoReal Analysis Exchange vol 39 no 2 pp 261ndash304 2013

[4] M Deveau and H Teismann ldquoWould real analysis be completewithout the fundamental theorem of calculusrdquo Elemente derMathematik vol 70 no 4 pp 161ndash172 2015

[5] J FHall ldquoCompleteness of ordered fieldsrdquo httpsarxivorgabs11015652v1

[6] R Kantrowitz and M Neumann ldquoAnother face of theArchimedean propertyrdquo The College Mathematics Journal vol46 no 2 pp 139ndash141 2015

[7] R Kantrowitz and M Schramm ldquoSeries that converge abso-lutely but donrsquot convergerdquoThe College Mathematics Journal vol43 no 4 pp 331ndash333 2012

[8] O Riemenschneider ldquo37 elementare axiomatische Charakter-isierungen des reellen ZahlkorpersrdquoMitteilungen derMathema-tischen Gesellschaft in Hamburg vol 20 pp 71ndash95 2001

[9] H Teismann ldquoToward a more complete list of completenessaxiomsrdquo American Mathematical Monthly vol 120 no 2 pp99ndash114 2013

[10] G Cantor ldquoUber unendliche lineare Punktmannichfaltig-keitenrdquoMathematische Annalen vol 21 no 4 pp 545ndash591 1883

[11] B Bollobas Linear Analysis An Introductory Course Cam-bridge University Press Cambridge UK 1990

[12] G L Cohen ldquoIs every absolutely convergent series convergentrdquoTheMathematical Gazette vol 61 no 417 pp 204ndash213 1977

[13] T M Apostol Mathematical Analysis Addison-Wesley Read-ing Mass USA 2nd edition 1974

[14] K Knopp Infinite Sequences and Series Dover PublicationsNew York NY USA 1956

[15] R Kantrowitz and M M Neumann ldquoMore of Dedekindhis series test in normed spacesrdquo International Journal ofMathematics and Mathematical Sciences vol 2016 Article ID2508172 3 pages 2016

[16] N Dunford and J T Schwartz Linear Operators Part I GeneralTheory John Wiley amp Sons New York NY USA 1958

[17] E Maor e The Story of a Number Princeton University PressPrinceton NJ USA 1994

[18] R G Bartle and D S Sherbert Introduction to Real AnalysisJohn Wiley amp Sons New York NY USA 3rd edition 2000

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the Archimedean property Other familiar tests that enter thefray are given as follows

Absolute Convergence Test If (119886119899) is a sequence of elements inthe ordered field F for which the seriessum |119886119899| converges thenthe series sum119886119899 also convergesComparisonTest If (119886119899) and (119887119899) are sequences of nonnegativeelements in the ordered field F that satisfy 119886119899 le 119887119899 for all 119899 andsum119887119899 converges then sum119886119899 converges

We document the connection between completeness andthese two series tests in the following theorem which maybe viewed as a modest complement to the main result of [2]Theorem 1 will be an important tool in this article

Theorem1 For an ordered field F the following statements areequivalent

(a) F is Cauchy complete(b) The absolute convergence test holds(c) The comparison test holds

Proof The fact that the absolute convergence test character-izes Cauchy completeness for ordered fields is proven in thearticle [2]

The validity of the comparison test in fields that areCauchy complete follows by noting that convergence of theseries sum119887119899 implies that its sequence (119861119899) of partial sums isconvergent and hence Cauchy The sequence (119860119899) of partialsums of the series sum119886119899 is therefore also Cauchy from whichthe convergence of the series sum119886119899 follows from the Cauchycompleteness of the field

Finally to see that the absolute convergence test holdswhenever the comparison test does observe that for all 119899the inequalities

0 le 119886119899 + 10038161003816100381610038161198861198991003816100381610038161003816 le 2 10038161003816100381610038161198861198991003816100381610038161003816 (1)

are true in F setting the stage for the comparison test toensure that the seriessum(119886119899+|119886119899|) converges Since alsosumminus|119886119899|converges it follows that sum119886119899 converges

Theorem 1 refines a result of [8] on the characteriza-tion of Dedekind complete ordered fields in terms of theArchimedean property together with either (b) or (c)

The equivalence of statements (a) and (b) of Theorem 1appears mutatis mutandis in functional analysis where itis known that a normed linear space is a Banach space ifand only if the absolute convergence test for series is valid(Theorem 28 of [11] or Statement (VIII) of [12])

Our goal in the present article is to involve the series testsof Dirichlet Dedekind and Abel in the taxonomy of orderedfieldsThe classical versions of these may be found for exam-ple in [13] or [14] but each has meaning in the more generalabstract context We devote Section 2 to their statements anda short discussion of them Like the absolute convergencetest and the comparison test Dirichletrsquos Dedekindrsquos andAbelrsquos tests are connected with completeness in the sense ofboth Dedekind and Cauchy These relationships are detailed

in Sections 4 and 5 Along the way we explore the role ofsequences of bounded variation in ordered fields

2 The Series Tests of DirichletDedekind and Abel

The hypothesis of Dedekindrsquos test calls for the sequence (119887119899)to have bounded variationwhich in the setting of the orderedfield F means that there is an element 119872 isin F for which theinequality sum119899119896=1 |119887119896+1 minus 119887119896| le 119872 holds for all 119899 isin N We willtreat sequences of bounded variation in some detail in thenext section Without further ado here is the trio of seriestests under consideration for an arbitrary ordered field

Dirichletrsquos Test If sum119886119899 is a series whose partial sums forma bounded sequence and (119887119899) is a decreasing sequence thatconverges to 0 then the seriessum119886119899119887119899 convergesDedekindrsquos Test If sum119886119899 is a convergent series and (119887119899) is asequence of bounded variation then the seriessum119886119899119887119899 conver-ges

Abelrsquos Test Ifsum119886119899 is a convergent series and (119887119899) is amonotoneconvergent sequence then the series sum119886119899119887119899 converges

Dirichletrsquos test is recognizable as a generalization ofLeibnizrsquos alternating series test from calculus (Theorem 342of [14]) and as Apostol mentions in Section 815 of [13] thesethree tests are useful tools when one is confronted with thetask of trying to determine convergence of real series thatdo not converge absolutely Moreover as Knopp points outin Section 55 of [14] a common feature of the tests in thistrio is that they all allow conclusions to be drawn about theseries sum119886119899119887119899 from assumptions concerning the series sum119886119899and the sequence (119887119899) Knopp remarks that such tests may beconstrued as ldquocomparison tests in the extended senserdquo andhe also includes a few applications and examples of them

It turns out that Abelrsquos test can be inferred fromDedekindrsquos test since as explained in the next section everymonotone convergent sequence has bounded variation Amomentrsquos reflection reveals that Abelrsquos test is also a corollaryof Dirichletrsquos test Indeed the sequence of partial sums of aconvergent series sum119886119899 is bounded so attention needs onlyto be directed to the stipulation that (119887119899) is a monotoneconvergent sequence If (119887119899) is decreasing and its limit is 119887then the auxiliary sequence given by 119888119899 = 119887119899 minus 119887 is alsodecreasing and it converges to 0 Dirichletrsquos test thus ensuresthat the seriessum119886119899119888119899 convergesThe convergence of the seriessum119886119899119887119899 follows easily from algebraic properties of convergentseries The case that (119887119899) is increasing with limit 119887 is handledanalogously since this time the sequence (minus119888119899) decreases to 0

None of the series tests of Dedekind Dirichlet or Abel ison its own strong enough to imply Dedekind completenessfor an ordered field F in which it holds If F is Archimedeanhowever and any of these three tests is in force then it willfollow from Theorem 3 that F is Dedekind complete Thelinchpin is a contractive-type property for certain sequencesin Archimedean ordered fields which is detailed in Lemma 4






119899

sum119896=1

1003816100381610038161003816119886119896+1 minus 1198861198961003816100381610038161003816 =119899

sum119896=1

(119886119896+1 minus 119886119896) = 119886119899+1 minus 1198861

le 1003816100381610038161003816119886119899+11003816100381610038161003816 minus 1198861 le 119870 minus 1198861(2)



le 100381610038161003816100381611988611003816100381610038161003816 + 119872(3)


1 0 12 013 0

14 0 (4)









119899

sum119896=1

119886119896119887119896 = 119860119899119887119899+1 minus119899

sum119896=1

119860119896 (119887119896+1 minus 119887119896) (5)



















119888119895 minus 119888119902 lt 119903 (119888119902 minus 119888119901) (7)




119888119895 minus 119888119899119896+1 lt 119903 (119888119899119896+1 minus 119888119899119896) (9)






119888119899119896+2 minus 119888119899119896+1 lt14 (119888119899119896+1 minus 119888119899119896) (10)


119886119896 = 12119896

119887119896 = 2119896 (119888119899119896+1 minus 119888119899119896)(11)


infin

sum119896=1

119886119896 =infin

sum119896=1

(12)119896

(12)





119898

sum119896=1

119886119896119887119896 =119898

sum119896=1

(119888119899119896+1 minus 119888119899119896) = 119888119899119898+1 minus 1198881198991 (14)





119909 + 11990922 + 1199093













10038161003816100381610038161003816(16)

and observe that

10038161003816100381610038161198861198961003816100381610038161003816 le10038161003816100381610038161003816119909119899119896


10038161003816100381610038161003816le 119899119896+1 minus 119899119896 (17)


119899119898+1minus1

sum119896=1


sum119896=1

119886119896 1003816100381610038161003816100381611990911989911989610038161003816100381610038161003816 (18)




Competing Interests


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119899

sum119896=1

1003816100381610038161003816119886119896+1 minus 1198861198961003816100381610038161003816 =119899

sum119896=1

(119886119896+1 minus 119886119896) = 119886119899+1 minus 1198861

le 1003816100381610038161003816119886119899+11003816100381610038161003816 minus 1198861 le 119870 minus 1198861(2)



le 100381610038161003816100381611988611003816100381610038161003816 + 119872(3)


1 0 12 013 0

14 0 (4)









119899

sum119896=1

119886119896119887119896 = 119860119899119887119899+1 minus119899

sum119896=1

119860119896 (119887119896+1 minus 119887119896) (5)



















119888119895 minus 119888119902 lt 119903 (119888119902 minus 119888119901) (7)




119888119895 minus 119888119899119896+1 lt 119903 (119888119899119896+1 minus 119888119899119896) (9)






119888119899119896+2 minus 119888119899119896+1 lt14 (119888119899119896+1 minus 119888119899119896) (10)


119886119896 = 12119896

119887119896 = 2119896 (119888119899119896+1 minus 119888119899119896)(11)


infin

sum119896=1

119886119896 =infin

sum119896=1

(12)119896

(12)





119898

sum119896=1

119886119896119887119896 =119898

sum119896=1

(119888119899119896+1 minus 119888119899119896) = 119888119899119898+1 minus 1198881198991 (14)





119909 + 11990922 + 1199093













10038161003816100381610038161003816(16)

and observe that

10038161003816100381610038161198861198961003816100381610038161003816 le10038161003816100381610038161003816119909119899119896


10038161003816100381610038161003816le 119899119896+1 minus 119899119896 (17)


119899119898+1minus1

sum119896=1


sum119896=1

119886119896 1003816100381610038161003816100381611990911989911989610038161003816100381610038161003816 (18)




Competing Interests


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























119888119895 minus 119888119902 lt 119903 (119888119902 minus 119888119901) (7)




119888119895 minus 119888119899119896+1 lt 119903 (119888119899119896+1 minus 119888119899119896) (9)






119888119899119896+2 minus 119888119899119896+1 lt14 (119888119899119896+1 minus 119888119899119896) (10)


119886119896 = 12119896

119887119896 = 2119896 (119888119899119896+1 minus 119888119899119896)(11)


infin

sum119896=1

119886119896 =infin

sum119896=1

(12)119896

(12)





119898

sum119896=1

119886119896119887119896 =119898

sum119896=1

(119888119899119896+1 minus 119888119899119896) = 119888119899119898+1 minus 1198881198991 (14)





119909 + 11990922 + 1199093













10038161003816100381610038161003816(16)

and observe that

10038161003816100381610038161198861198961003816100381610038161003816 le10038161003816100381610038161003816119909119899119896


10038161003816100381610038161003816le 119899119896+1 minus 119899119896 (17)


119899119898+1minus1

sum119896=1


sum119896=1

119886119896 1003816100381610038161003816100381611990911989911989610038161003816100381610038161003816 (18)




Competing Interests


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119898

sum119896=1

119886119896119887119896 =119898

sum119896=1

(119888119899119896+1 minus 119888119899119896) = 119888119899119898+1 minus 1198881198991 (14)





119909 + 11990922 + 1199093













10038161003816100381610038161003816(16)

and observe that

10038161003816100381610038161198861198961003816100381610038161003816 le10038161003816100381610038161003816119909119899119896


10038161003816100381610038161003816le 119899119896+1 minus 119899119896 (17)


119899119898+1minus1

sum119896=1


sum119896=1

119886119896 1003816100381610038161003816100381611990911989911989610038161003816100381610038161003816 (18)




Competing Interests


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Competing Interests


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Completeness of Ordered Fields and a Trio ...downloads.hindawi.com › journals › aaa › 2016 › 6023273.pdf · Research Article Completeness of Ordered Fields