# Double Sequences and Selections

Authors: Dragan Djurčić; Ljubiša D. R. Kočinac; Mališa R. Žižović
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We present some results about selection properties in the class of double sequences of real numbers.


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## Body

## 1. Introduction

In 1900, Pringsheim introduced the concept of convergence of real double sequences: a double sequenceX=(xm,n)m,n∈ℕ converges to a∈ℝ (notation P-limX=a or P-limxm,n=a), if for every ε>0 there is n0∈ℕ such that |xm,n-a|<ε for all m,n>n0 (see [1], and also [2, 3]). The limit a is called the Pringsheim limit of X.In this paper we denote byc2a the set of all double real sequences converging to a point a∈ℝ in Pringsheim's sense.A considerable number of papers which appeared in recent years study the setc2a and its subsets from various points of view (see [4–13]). Some results in this investigation are generalizations of known results concerning simple sequences to certain classes of double sequences, while other results reflect a specific nature of the Pringsheim convergence (e.g., the fact that a double sequence may converge without being bounded). In this paper we begin with a quite different investigation of double sequences related to selection principles (and games corresponded to them): for a given sequence (Xn:n∈ℕ) of double sequences that belong to one class 𝒜 we select from each Xn a subset Yn by a prescribed procedure so that Yn's may be arranged to a new double sequence Y which belongs to another class ℬ(not necessarily distinct from 𝒜) of double sequences. (For selection principles theory see [14, 15], and for selection properties of some classes of simple sequences see [16–19]). Moreover, our investigation suggests also introduction of new selection principles: instead of a sequence of double sequences from a class 𝒜 we start with a double sequence of double sequences from 𝒜 (see Definitions 2.1 and 2.7). The classes of double sequences considered in this article are subsets of the class c2a and will be defined below.IfP-lim|X|=∞, (equivalently, for every M>0 there are n1,n2∈ℕ such that |xm,n|>M whenever m≥n1, n≥n2), then X is said to be definitely divergent.A double sequenceX=(xm,n)m,n∈ℕ is bounded if there is M>0 such that |xm,n|<M for all m,n∈ℕ.Notice that aP-convergent double sequence need not be bounded.A numberL∈ℝ is said to be a Pringsheim limit point of a double sequence X=(xm,n)m,n∈ℕ if there exist two increasing sequences m1<m2⋯<mi,⋯ and n1<n2⋯<ni,⋯ such that
(1.1)limi→∞xmi,ni=L.In [20], Hardy introduced the notion of regular convergence for double sequences: a double sequence X=(xm,n)m,n∈ℕ regularly converges to a point a if it P-converges to a and for each m∈ℕ and each n∈ℕ there exist the following two limits:
(1.2)limn→∞xm,n=Rm,limm→∞xm,n=Cn.The symbolc¯2a denotes the set of elements (xm,n)m,n∈ℕ in c2a which are bounded, regular and such that limm→∞xm,n=limn→∞xm,n=a.


## 2. Results

We begin with the following new selection principle for classes of double sequences.Definition 2.1.
Let𝒜 and ℬ be subclasses of c2a. Then S1(d)(𝒜,ℬ) denotes the selection hypothesis: for each double sequence (Am,n:m,n∈ℕ) of elements of 𝒜 there are elements am,n∈Am,n such that the double sequence (am,n)m,n∈ℕ belongs to ℬ.Theorem 2.2.
Fora∈ℝ the selection principle S1(d)(c2a,c2a) is true.Proof.
Let(Xj,k:j,k∈ℕ) be a double sequence of elements in c2a. Suppose that Xj,k=(xm,nj,k)m,n∈ℕ for all j,k∈ℕ. Let us construct a double sequence Y=(ym,n)m,n∈ℕ in the following way:(1)
y1,1=xm1,m11,1∈X1,1, where m1∈ℕ is such that |xm,n1,1-a|≤1/2 for each m≥m1 and each n≥m1.(2)
y1,2=xm2,m21,2∈X1,2, where m2∈ℕ is such that |xm,n1,2-a|≤1/22 for each m≥m2 and each n≥m2.(3)
y2,1=xm3,m32,1∈X2,1 with m3∈ℕ such that |xm,n2,1-a|≤1/22 for each m≥m3 and each n≥m3.(4)
y2,2=xm4,m42,2∈X2,2 with m4∈ℕ such that |xm,n2,2-a|≤1/22 for each m≥m4 and each n≥m4.
In general, fors,t∈ℕ, q=max{s,t}≥3, we put ys,t=xmp,mps,t, where
(2.1)p={(q-1)2+t,ifq=s,(q-1)2+2t-s,ifq=t,
and |xm,ns,t-a|≤1/2q for each m≥mp and each n≥mp.
We prove that the double sequenceY=(ym,n)m,n∈ℕ∈c2a. Let ε>0 be given. Pick r∈ℕ such that 1/2r<ε. For each m≥r and each n≥r, by construction of Y, we have |ym,n-a|≤1/2r<ε. This just means Y∈c2a. As ym,n∈Xm,n for all m,n∈ℕ, the theorem is proved.Remark 2.3.
The double sequenceY from the proof of Theorem 2.2 has also the following properties: (i) Y is bounded; (ii) Y is regular and limm→∞ym,n=limn→∞ym,n=a for each m∈ℕ and each n∈ℕ, that is  Y∈c¯2a.Definition 2.4 (see [15]).
Let𝒜 and ℬ be subclasses of c2a. Then α2(𝒜,ℬ) denotes the selection hypothesis: for each sequence (An:n∈ℕ) of elements of 𝒜 there is an element B in ℬ such that B∩An is infinite for all n∈ℕ.Lemma 2.5.
Fora∈ℝ, the selection principle α2(c2a,c2a) is satisfied.Proof.
Let(Sk:k∈ℕ) be a sequence of elements from c2a and let for each k∈ℕ,  Sk=(xm,nk)m,n∈ℕ.(1)
Form first an increasing sequencej1<j2<⋯<ji<⋯ in ℕ so that:
(a)
j1=min{n0∈ℕ:|xm,n1-a|≤1/2forallm≥n0andforalln≥n0};(b)
Leti≥2. Find pi=min{n0∈ℕ:|xm,ni-a|≤1/2iforallm,n≥n0}, and then define
(2.2)ji={pi,ifpi>ji-1;ji-1+1,ifpi≤ji-1.(2)
Define now a double sequenceY=(ys,t)s,t∈ℕ in this way:
(a)
ys,t=xs,t1 for each 1≤s<j2, t∈ℕ, and each 1≤t<j2, s∈ℕ;(b)
fori≥2,  ys,t=xs,ti, for ji≤s<ji+1, t≥ji, and ji≤t<ji+1, s≥ji.By construction, Y∈c2a and Y has infinitely many common elements with each Xk, k∈ℕ; that is, the selection principle α2(c2a,c2a) is satisfied.Remark 2.6.
Using the technique from [17] we can prove that the double sequence Y in the proof of the previous lemma can be chosen in such a way that Y has infinitely many common elements with each  Xk, k∈ℕ, but on the same (corresponding) positions.
Let for eachk∈ℕ, xk denote the sequence (xm,mk)m∈ℕ. Then each xk converges to a, so that we have the sequence (xk:k∈ℕ) of sequences converging to a. Let 2=p1<p2<p3<⋯ be a sequence of prime natural numbers. Take sequence x1=(xm,m1)m∈ℕ. For each i∈ℕ, replace the elements of x1 on the positions pih, h∈ℕ, by the corresponding elements of the sequence xi+1. One obtains the sequence (zm)m∈ℕ converging to a which has infinitely many common elements with each xk on the same positions as in xk. Define now the double sequence Y=(ys,t)s,t∈ℕ so that ys,s=zs, s∈ℕ, and ys,t=a whenever s≠t. By construction, Y∈c¯2a and has infinitely many common positions with each  Xk.The following definition gives a double sequence version of the selection propertyα2(𝒜,ℬ).Definition 2.7.
Let𝒜 and ℬ be subclasses of c2a. Then α2(d)(𝒜,ℬ) denotes the selection hypothesis: for each double sequence (Am,n:m,n∈ℕ) of elements of 𝒜 there is an element B in ℬ such that B∩Am,n is infinite for all (m,n)∈ℕ×ℕ.Theorem 2.8.
Leta∈ℝ be given. The selection principle α2(d)(c2a,c2a) is true.Proof.
Let(Xj,k:j,k∈ℕ) be a double sequence of elements in c2a and let Xj,k=(xm,nj,k)m,n∈ℕ. In a standard way (see [2]) form from this double sequence a sequence (Xi:i∈ℕ) of double sequences Xi=(xm,ni)m,n∈ℕ. Apply now Lemma 2.5 to this sequence and find a double sequence Y∈c2a such that Y∩Xi is infinite for each i∈ℕ. But then Y∩Xj,k is infinite for all j,k∈ℕ.Remark 2.9.
Notice that the double sequenceY from the proofs of Lemma 2.5 and Theorem 2.8 satisfies: (a) Y is bounded; (b) Y is regular, and limn→∞ym,n=limm→∞ym,n=a for each m∈ℕ and each n∈ℕ.Theorem 2.10.
Leta∈ℝ and let (Xk:k∈ℕ) be a sequence of double sequences in c2a, Xk=(xm,nk)m,n∈ℕ. Then there is a double sequence Y=(ys,t)s,t∈ℕ in c2a such that for each k∈ℕ the set {(s,t)∈ℕ×ℕ:ys,t=xm,nkfor  some(m,n)∈ℕ×ℕ} is infinite.Proof.
The double sequenceY is defined in the following way.
Letk∈ℕ. There is ik∈ℕ such that |xm,nk-a|<2-k for all m,n≥ik. Let
(2.3)s*={ik,fors=k,ik+p,fors=k+p,p∈ℕ,t*={ik,fort=k,ik+p,fort=k+p,p∈ℕ.
For t≥k let yk,t=xik,t*k, and for s≥k let ys,k=xs*,ikk. The double sequence Y=(ys,t)s,t∈ℕ constructed in this way is as required, because Y has the following properties:(1)
Y∈c2a;(2)
the setBk={yk,t:t≥k}∪{ys,k:s≥k} is a subset of Ak={xm,nk:m,n∈ℕ};(3)
for eachk∈ℕ, Bk is countable;(4)
⋃k∈ℕ‍Bk={ys,t:s,t∈ℕ}.Another similar result is given in the next theorem.Theorem 2.11.
Leta∈ℝ and let (Xk:k∈ℕ) be a sequence of double sequences in c2a, Xk=(xm,nk)m,n∈ℕ. Then there is a double sequence Y=(ys,t)s,t∈ℕ in c2a which has one common row with Xk for each  k∈ℕ.Proof.
For eachk∈ℕ there is n0(k)∈ℕ such that |xm,nk-a|<2-k for all m,n≥n0(k), n0(k1)>n0(k2) whenever k1>k2, and n0(k)≥min{i(k)∈ℕ:|xm,nk+1-a|<2-kfor  allm,n≥i(k)}. Then the desired double sequence Y is defined in such a way that its n0(k)th row is the n0(k)th row of Xk, that is yn0(k),n=xn0(k),nk (n∈ℕ), and ys,t=a otherwise. Let us prove that Y∈c2a. Indeed, if ε>0 is given, then choose p∈ℕ such that 2-p<ε. Then for each k∈ℕ we have |xm,nk-a|<ε for all m,n≥p. By construction of Y we have actually that |ym,n-a|<ε for all m,n≥p, that is  Y∈c2a.Consider now an order on the setℕ×ℕ. Let φ:ℕ×ℕ→ℕ be a bijection. Set (m1,n1)≤φ(m2,n2)⇔φ(m1,n1)≤φ(m2,n2), where ≤ is the natural order in ℕ.Definition 2.12.
Let𝒜 and ℬ be subclasses of c2a. Then S1φ(𝒜,ℬ) denotes the selection hypothesis: for each sequence (An:n∈ℕ) of elements of 𝒜 there is an element B=(bφ-1(n))n∈ℕ in ℬ such that bφ-1(n)∈An for all n∈ℕ.Theorem 2.13.
Leta∈ℝ and let ≤φ be as previously mentioned. Then the selection hypothesis S1φ(c2a,c¯2a) is satisfied.Proof.
Let(Xk:k∈ℕ), Xk=(xm,nk)m,n∈ℕ, be a sequence in c2a. Construct a double sequence Y=(ys,t)s,t∈ℕ as follows.
Fixk∈ℕ. Let (s(k),t(k))=φ-1(k), and let p(k)=max{s(k),t(k)}. There is n0(k)∈ℕ such that |xm,nk-a|<2-p(k) for all m,n≥n0(k). Set ys(k),t(k)=xn0(k),n0(k)k and Y=(ys(k),t(k))k∈ℕ. Then, by the construction, Y∈c¯2a and Y have exactly one common element with Xk for each k∈ℕ, that is  Y is the desired selector.


## 3. Concluding Remarks

We considered here selection properties of some classes of convergent double sequences. It would be interesting also to study similar properties for classes of divergent double sequences, as well as selections related to the Pringsheim limit points instead of theP-limits.



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