# Summability of Sequences and Selection Properties

Authors: Dragan Djurčić; Ljubiša D. R. Kočinac; Mališa R. Žižović
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We prove that some classes of summable sequences of positive real numbers satisfy several selection principles related to a special kind of convergence.


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## Body

## 1. Introduction

Byℕ, ℝ, and ℝ¯ we denote the set of natural numbers, real numbers, and the extended real line ℝ∪{-∞,∞}, respectively.Let𝕊 denote the set of sequences a=(an)n∈ℕ of positive real numbers.We begin with the following definitions of selection principles.Let𝒜 and ℬ be nonempty subsets of 𝕊. Then the symbol S1(𝒜,ℬ) denotes the selection principle.For each sequence(an:n∈ℕ) of elements of 𝒜 there is a sequence b=(bn)n∈ℕ∈ℬ such that bn∈an for each n∈ℕ.The following infinitely long gameG1(𝒜,ℬ) is naturally associated to the previous selection principle. Two players, ONE and TWO, play a round for each positive integer. In then-th round ONE chooses a sequence an∈𝒜, and TWO responds by choosing an element bn∈an. TWO wins a play (a1,b1;…;an,bn;…) if b=(bn)n∈ℕ∈ℬ; otherwise, ONE wins.Another selection principleSfin(𝒜,ℬ) is defined as follows.For each sequence(an:n∈ℕ) of elements of 𝒜 there is a sequence b∈ℬ such that b∩an is finite for each n∈ℕ.It is clear how the corresponding gameGfin(𝒜,ℬ) is defined.A strategy of a player is a functionσ from the set of all finite sequences of moves of the other player into the set of admissible moves of the strategy owner.A strategyσ for the player TWO is a coding strategy if TWO remembers only the most recent move by ONE and by TWO before his next move. More precisely the moves of TWO are b1=σ(a1,∅); bn=σ(an,bn-1), n≥2.In this paper we introduce also the following game. Leti∈ℕ be a fixed (but arbitrary) natural number. We define the game G1(w=i)(𝒜,ℬ) for two players, ONE and TWO, who play a round for each n∈ℕ. In the i-th round ONE plays a sequence ai=(ai,m)m∈ℕ∈𝒜, and TWO responds by choosing a finite set Fi={ai,mi1,…,ai,mik}. In the n-th round, n≠i, ONE plays a sequence an=(an,m)m∈ℕ∈𝒜, and TWO responds by choosing an element an,mn∈an. TWO wins a play if the sequence b=(a1,m1,…,ai-1,mi-1;ai,mi1,…,ai,mik;ai+1,mi+1,…) belongs to ℬ; otherwise, ONE wins.For more information on selection principles and games see the survey papers in [1, 2] and references therein.In a number of papers by the authors published in the last few years it was demonstrated that some subclasses𝒜 and ℬ of 𝕊 satisfy certain selection principles and game theoretical statements (for 𝒜 and ℬ classes of divergent sequences related to celebrated Karamata's theory of regular variation [3–6] see [7–12], and for 𝒜 and ℬ classes of sequences converging to 0 see [13]). For other results concerning sequences and sequence spaces see [14–16].In this paper our selections are related to special kinds of convergence of series. More precisely, we start by a sequence of summable sequences and during the selection process we control not only the convergence of series, but also the nature of that convergence.


## 2. Results

We use the following notations for the classes of sequences we deal with:(2.1)l1={a∈S:∑n=1∞an<∞},l1,S={a∈S:∑n=1∞an=S},forS∈(0,∞],l1,(α,β)={a∈l1,S:S∈(α,β)},forα,β∈(0,∞),l1,(α,β]=l1,(α,β)⋃l1,β,forα,β∈(0,∞).Notice that the sequencex=(xn)n∈ℕ, xn=S/2n, belongs to the class ℓ1,S, so that all the classes above are nonempty.Theorem 2.1.
For eachS∈(0,∞) and each ɛ=ɛ(S)∈(0,S)  TWO has a winning coding strategy in the game G1(w=1)(ℓ1,S,ℓ1,(S-ɛ,S]).Proof.
Letσ denote a strategy of TWO, and let S>0 and ɛ=ɛ(S)∈(0,S) be fixed. Suppose that in the first round ONE chooses a sequence x1=(x1,m)m∈ℕ from ℓ1,S. There is k∈ℕ such that ∑m=k+1∞x1,m<ɛ/2, and thus M=S-∑m=1kx1,m∈(0,ɛ/2). Player TWO plays σ(x1)={x1,1,…,x1,k}—a finite subset of x1.
In the second round ONE chooses a sequencex2=(x2,m)m∈ℕ∈ℓ1,S, and then TWO responds by choosing σ(x2,σ(x1))=x2,m2 such that x2,m2<M/2 (which is possible because limm→∞x2,m=0).
In then-th round, n≥3, ONE chooses xn=(xn,m)m∈ℕ∈ℓ1,S, and TWO's response is σ(xn,xn-1,mn-1)=xn,mn such that xn,mn<xn-1,mn-1/2n-1<M/2n-1, and so on.
Setyn=x1,n for n≤k and yn=xn-k+1,mn-k+1 for n>k. Let us prove y=(yn)n∈ℕ∈ℓ1,(S-ɛ,S]. We have
(2.2)∑n=1∞yn=∑n=1kyn+∑n=k+1∞yn=∑m=1kx1,m+∑n=k+1∞yn=S-M+∑n=k+1∞yn<S-M+M(∑i=1∞12i)=S.
On the other hand,
(2.3)∑n=1∞yn>∑m=1kx1,m=S-M>S-ɛ2.
That is, y∈ℓ1,(S-ɛ,S].Corollary 2.2.
For eachS∈(0,∞) and each ɛ=ɛ(S)∈(0,S) the selection principle Sfin(ℓ1,S,ℓ1,(S-ɛ,S]) is true.Notice that one can prove a refinement of Theorem2.1 (and Corollary 2.2) in the sense that it is possible to have additional control of selections giving the sequence y. For this we need the following definitions and notation.Definition 2.3 (see [13]).
A sequence(xn)n∈ℕ∈𝕊 is said to belong to the class Tr(R-∞,s) if for each λ≥1 it satisfies
(2.4)lim⁡n→∞x[n+λ]xn=0,
where [r] denotes the integer part of r∈ℝ.Definition 2.4 (see [9]).
For a sequencex=(xn)n∈ℕ∈𝕊, the Landau-Hurwicz sequencew(x)=(wn(x))n∈ℕ of x is defined by
(2.5)wn(x):=sup⁡{|xm-xk|:m≥n,k≥n},n∈N.Given a sequencex=(xn)n∈ℕ∈𝕊 we denote by Sx=(Sn(x))n∈ℕ the sequence defined by(2.6)Sn(x)=∑i=1nxi,n∈N.LetℓTr(R-∞,s)1,(α,β] be the set of all sequences a=(an)n∈ℕ∈ℓ1,(α,β] such that w(Sa)∈Tr(R-∞,s).Theorem 2.5.
For eachS∈(0,∞) and each ɛ=ɛ(S)∈(0,S)  TWO has a winning coding strategy in the game G1(w=1)(ℓ1,S,ℓTr(R-∞,s)1,(S-ɛ,S]).Proof.
The strategyσ of player TWO and the sequence y=(yn)n∈ℕ are actually from the proof of Theorem 2.1. Therefore, y∈ℓ1,(S-ɛ,S]. Besides, since, by construction, the series
(2.7)∑n=1∞yn+1yn
is convergent, we have
(2.8)lim⁡n→∞(∑k=n∞yk+1yk)=0.
Consider now the sequenceSy=(Sn(y))n∈ℕ. This sequence is convergent (by the d'Alembert criterion), and let S(y) be its limit. It remains to prove w(Sy)=(wn(Sy))n∈ℕ∈Tr(R-∞,s). It is enough to prove
(2.9)lim⁡n→∞wn+1(Sy)wn(Sy)=0.
First, notice that(2.10)wn(Sy)=S(y)-Sn(y),n∈N.
Thus we get
(2.11)lim⁡n→∞wn+1(Sy)wn(Sy)=lim⁡n→∞S(y)-Sn+1(y)S(y)-Sn(y)=1-lim⁡n→∞yn+1yn+1+yn+2+⋯=0.
That is (2.9), since by (2.8) and the fact that for n sufficiently large it holds
(2.12)yn+2yn+1+yn+3yn+1+⋯=yn+2yn+1+yn+3yn+2⋅yn+2yn+1+⋯≤yn+2yn+1+yn+3yn+2+⋯,
we have
(2.13)lim⁡n→∞yn+1yn+1+yn+2+⋯=lim⁡n→∞11+(yn+2/yn+1)+(yn+3/yn+1)+⋯=1.
The theorem is proved.Corollary 2.6.
The selection principleSfin(ℓ1,S,ℓTr(R-∞,s)1,(S-ɛ,S]) is true.The following two theorems give other selection results for defined classes of sequences: one of theSfin-type and the other of the S1-type.Theorem 2.7.
For eachS∈(0,∞] the selection principle Sfin(ℓ1,S,ℓ1,∞) is satisfied.Proof.
Consider first the caseS∈(0,∞). Let (xn:n∈ℕ), xn=(xn,m)m∈ℕ, be a sequence of elements of ℓ1,S. For each n∈ℕ let zni=xn,i, i≤k=k(n), be a finite subset of xn such that S/2<∑i=1kzni<S. Arrange now znp, n∈ℕ, p∈{1,2,…,k(n)}, in the sequence y=(yj)j∈ℕ in which the position of an element is determined first by n and then by p, that is,
(2.14)y=(z11,…,z1k(1);…;zn1,…,znk(n);…).
We have
(2.15)n⋅S2<∑m=1n∑i=1k(m)zmi=∑j=1k(n)yj,
where yk(n) is the last element of xn belonging to the sequence y. Therefore,
(2.16)∑j=1∞yj=lim⁡n→∞∑j=1k(n)yj>lim⁡n→∞(n⋅S2)=∞.
That is, y∈ℓ1,∞.
Suppose now thatS=∞. This case is treated similarly to the previous case, but here we require ∑i=1kzni>1 for each n∈ℕ; the sequence y=(yj)j∈ℕ is formed in a similar way as in the first case. So we have n·1<∑j=1k(n)yj for each n∈ℕ, hence
(2.17)∑j=1∞yj=lim⁡n→∞∑j=1k(n)yj>lim⁡n→∞(n⋅1)=∞.
That is, y∈ℓ1,∞. The theorem is proved.Theorem 2.8.
For eachS∈(0,∞) and each α>0 the selection principle S1(ℓ1,S,ℓ1,(0,α)) is true.Proof.
Let(xn:n∈ℕ), xn=(xn,m)m∈ℕ, be a sequence of elements in ℓ1,S. For each n∈ℕ take yn=xn,mn∈xn so that y1∈(0,α) (which is possible since x1,m→0 as m→∞) and yn<α-y1/2n-1 for n≥2. Then the sequence y=(yn)n∈ℕ witnesses that the statement is true, because
(2.18)∑n=1∞yn=y1+∑n=2∞yn<y1+(α-y1)=α.
That is, y∈ℓ1,(0,α).For the next result we have to define the following selection principles [2, 17]. Notice that in [18] we developed an interesting technique for proving results concerning these selection principles and certain classes of sequences from 𝕊. In [19] we proposed the use of this technique (and these selection principles) in other fields of mathematics and its applications.Let, as before,𝒜 and ℬ be certain nonempty subfamilies of 𝕊. Then the symbol αi(𝒜,ℬ), i=2,3,4, denotes the selection hypothesis that for each sequence (an:n∈ℕ) of elements of 𝒜 there is an element b∈ℬ such that:α2(A,B): for each n∈ℕ the set an∩b is infinite;α3(𝒜,ℬ): for infinitely many n∈ℕ the set an∩b is infinite;α4(𝒜,ℬ): for infinitely many n∈ℕ the set an∩b is nonempty.Theorem 2.9.
For eachS∈(0,∞) and each α>0 the selection principles αi(ℓ1,S,ℓTr(R-∞,s)1,(0,α)), i=2,3,4, are satisfied.Proof.
We prove that the principleα2 is true (hence also α3 and α4). Let (xn:n∈ℕ), xn=(xn,m)m∈ℕ, be a sequence of sequences from ℓ1,S. Let m1∈ℕ be such that ∑m=m1+1∞x1,m<α/2. For k≤2 let mk be a natural number such that ∑m=mk+1∞xk,m<α/2k. Consider the sequence y=(yj)j∈ℕ defined in this way:
(2.19)y=(x1,m1+1,x1,m2+2,…;x2,m2+1,x2,m2+2,…;xk,mk+1,xk,mk+2,…).
Then y∩xn is infinite for each n∈ℕ. Further, y∈ℓ1,(0,α) because
(2.20)0<∑j=1∞yj=∑k=1∞∑m=mk+1∞xk,m<∑k=1∞α2k=α.
We construct now a new sequencez=(zi) in the way  described in Table 1.
Evidently,z∩xn is infinite for each n∈ℕ. Also, 0<∑i=1∞zi≤∑j=1∞yj<α, that is, z∈ℓ1,(0,α). By a minor modification of the proof of Theorem 2.5 we obtain w(Sz)∈Tr(R-∞,s). This means z∈ℓTr(R-∞,s)1,(0,α).Table 1x1x2x3x4Howz1z1∈y∩x1———anyz2—z2∈y∩x2——z2/z1<1/2z3z3∈y∩x1———z3/z2<1/22z4——z4∈y∩x3—z4/z3<1/23z5—z5∈y∩x2——z5/z4<1/24z6z6∈y∩x1———z6/z5<1/25z7———z7∈y∩x4z7/z6<1/26z8——z8∈y∩x3—z8/z7<1/27z9⋯⋯⋯⋯⋯



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