Double Sequences and Selections

Abstract

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

1. Introduction

In 1900, Pringsheim introduced the concept of convergence of real double sequences: a double sequence 𝑋=(𝑥_𝑚,𝑛)_{𝑚,𝑛∈ℕ} converges to 𝑎∈ℝ (notation 𝖯-lim𝑋=𝑎 or 𝖯-lim𝑥_𝑚,𝑛=𝑎), if for every 𝜀>0 there is 𝑛₀∈ℕ such that |𝑥_𝑚,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 by 𝑐^𝑎2 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 set 𝑐^𝑎2 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 (𝑋_𝑛∶𝑛∈ℕ) of double sequences that belong to one class 𝒜 we select from each 𝑋_𝑛 a subset 𝑌_𝑛 by a prescribed procedure so that 𝑌_𝑛'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 𝑐^𝑎2 and will be defined below.

If 𝖯-lim|𝑋|=∞, (equivalently, for every 𝑀>0 there are 𝑛₁,𝑛₂∈ℕ such that |𝑥_𝑚,𝑛|>𝑀 whenever 𝑚≥𝑛₁, 𝑛≥𝑛₂), then 𝑋 is said to be definitely divergent.

A double sequence 𝑋=(𝑥_𝑚,𝑛)_{𝑚,𝑛∈ℕ} is bounded if there is 𝑀>0 such that |𝑥_𝑚,𝑛|<𝑀 for all 𝑚,𝑛∈ℕ.

Notice that a 𝖯-convergent double sequence need not be bounded.

A number 𝐿∈ℝ is said to be a Pringsheim limit point of a double sequence 𝑋=(𝑥_𝑚,𝑛)_{𝑚,𝑛∈ℕ} if there exist two increasing sequences 𝑚₁<𝑚₂⋯<𝑚_𝑖,⋯ and 𝑛₁<𝑛₂⋯<𝑛_𝑖,⋯ such that lim_𝑖→∞𝑥_{𝑚𝑖,𝑛𝑖}=𝐿.(1.1)

In [20], Hardy introduced the notion of regular convergence for double sequences: a double sequence 𝑋=(𝑥_𝑚,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: lliimm_{𝑛→∞𝑚→∞}𝑥_𝑚,𝑛=𝑅_𝑚,𝑥_𝑚,𝑛=𝐶_𝑛.(1.2)

The symbol 𝑐^𝑎2 denotes the set of elements (𝑥_𝑚,𝑛)_{𝑚,𝑛∈ℕ} in 𝑐^𝑎2 which are bounded, regular and such that lim_𝑚→∞𝑥_𝑚,𝑛=lim_𝑛→∞𝑥_𝑚,𝑛=𝑎.

2. Results

We begin with the following new selection principle for classes of double sequences.

Definition 2.1. Let 𝒜 and ℬ be subclasses of 𝑐^𝑎2. Then 𝖲^1(𝖽)(𝒜,ℬ) denotes the selection hypothesis: for each double sequence (𝐴_𝑚,𝑛∶𝑚,𝑛∈ℕ) of elements of 𝒜 there are elements 𝑎_𝑚,𝑛∈𝐴_𝑚,𝑛 such that the double sequence (𝑎_𝑚,𝑛)_{𝑚,𝑛∈ℕ} belongs to ℬ.

Theorem 2.2. For 𝑎∈ℝ the selection principle 𝖲^1(𝖽)(𝑐^𝑎2,𝑐^𝑎2) is true.

Proof. Let (𝑋^𝑗,𝑘∶𝑗,𝑘∈ℕ) be a double sequence of elements in 𝑐^𝑎2. Suppose that 𝑋^𝑗,𝑘=(𝑥^{𝑗,𝑘𝑚,𝑛})_{𝑚,𝑛∈ℕ} for all 𝑗,𝑘∈ℕ. Let us construct a double sequence 𝑌=(𝑦_𝑚,𝑛)_{𝑚,𝑛∈ℕ} in the following way:(1)𝑦_1,1=𝑥^𝑚1,1_1,𝑚1∈𝑋^1,1, where 𝑚₁∈ℕ is such that |𝑥^1,1𝑚,𝑛−𝑎|≤1/2 for each 𝑚≥𝑚₁ and each 𝑛≥𝑚₁.(2)𝑦_1,2=𝑥^𝑚1,2_2,𝑚2∈𝑋^1,2, where 𝑚₂∈ℕ is such that |𝑥^1,2𝑚,𝑛−𝑎|≤1/2² for each 𝑚≥𝑚₂ and each 𝑛≥𝑚₂.(3)𝑦_2,1=𝑥^𝑚2,1_3,𝑚3∈𝑋^2,1 with 𝑚₃∈ℕ such that |𝑥^2,1𝑚,𝑛−𝑎|≤1/2² for each 𝑚≥𝑚₃ and each 𝑛≥𝑚₃.(4)𝑦_2,2=𝑥^𝑚2,2_4,𝑚4∈𝑋^2,2 with 𝑚₄∈ℕ such that |𝑥^2,2𝑚,𝑛−𝑎|≤1/2² for each 𝑚≥𝑚₄ and each 𝑛≥𝑚₄.
In general, for 𝑠,𝑡∈ℕ, 𝑞=max{𝑠,𝑡}≥3, we put 𝑦_𝑠,𝑡=𝑥^{𝑚𝑠,𝑡}_{𝑝,𝑚𝑝}, where 𝑝=(𝑞−1)²(𝑞−1)₂+𝑡,if𝑞=𝑠,+2𝑡−𝑠,if𝑞=𝑡,(2.1) and |𝑥^{𝑠,𝑡𝑚,𝑛}−𝑎|≤1/2^𝑞 for each 𝑚≥𝑚_𝑝 and each 𝑛≥𝑚_𝑝.
We prove that the double sequence 𝑌=(𝑦_𝑚,𝑛)_{𝑚,𝑛∈ℕ}∈𝑐^𝑎2. Let 𝜀>0 be given. Pick 𝑟∈ℕ such that 1/2^𝑟<𝜀. For each 𝑚≥𝑟 and each 𝑛≥𝑟, by construction of 𝑌, we have |𝑦_𝑚,𝑛−𝑎|≤1/2^𝑟<𝜀. This just means 𝑌∈𝑐^𝑎2. As 𝑦_𝑚,𝑛∈𝑋^𝑚,𝑛 for all 𝑚,𝑛∈ℕ, the theorem is proved.

Remark 2.3. The double sequence 𝑌 from the proof of Theorem 2.2 has also the following properties: (i) 𝑌 is bounded; (ii) 𝑌 is regular and lim_𝑚→∞𝑦_𝑚,𝑛=lim_𝑛→∞𝑦_𝑚,𝑛=𝑎 for each 𝑚∈ℕ and each 𝑛∈ℕ, that is 𝑌∈𝑐^𝑎2.

Definition 2.4 (see [15]). Let 𝒜 and ℬ be subclasses of 𝑐^𝑎2. Then 𝛼₂(𝒜,ℬ) denotes the selection hypothesis: for each sequence (𝐴_𝑛∶𝑛∈ℕ) of elements of 𝒜 there is an element 𝐵 in ℬ such that 𝐵∩𝐴_𝑛 is infinite for all 𝑛∈ℕ.

Lemma 2.5. For 𝑎∈ℝ, the selection principle 𝛼₂(𝑐^𝑎2,𝑐^𝑎2) is satisfied.

Proof. Let (𝑆^𝑘∶𝑘∈ℕ) be a sequence of elements from 𝑐^𝑎2 and let for each 𝑘∈ℕ, 𝑆^𝑘=(𝑥^{𝑘𝑚,𝑛})_{𝑚,𝑛∈ℕ}.(1)Form first an increasing sequence 𝑗₁<𝑗₂<⋯<𝑗_𝑖<⋯ in ℕ so that:(a)𝑗₁=min{𝑛₀∈ℕ∶|𝑥^1𝑚,𝑛−𝑎|≤1/2forall𝑚≥𝑛₀andforall𝑛≥𝑛₀};(b)Let 𝑖≥2. Find 𝑝_𝑖=min{𝑛₀∈ℕ∶|𝑥^{𝑖𝑚,𝑛}−𝑎|≤1/2^𝑖forall𝑚,𝑛≥𝑛₀}, and then define 𝑗_𝑖=𝑝^𝑖𝑗,_𝑖−1if𝑝_𝑖>𝑗_𝑖−1;+1,if𝑝_𝑖≤𝑗_𝑖−1.(2.2)(2)Define now a double sequence 𝑌=(𝑦_𝑠,𝑡)_{𝑠,𝑡∈ℕ} in this way:(a)𝑦_𝑠,𝑡=𝑥^1𝑠,𝑡 for each 1≤𝑠<𝑗₂, 𝑡∈ℕ, and each 1≤𝑡<𝑗₂, 𝑠∈ℕ;(b)for 𝑖≥2, 𝑦_𝑠,𝑡=𝑥^{𝑖𝑠,𝑡}, for 𝑗_𝑖≤𝑠<𝑗_𝑖+1, 𝑡≥𝑗_𝑖, and 𝑗_𝑖≤𝑡<𝑗_𝑖+1, 𝑠≥𝑗_𝑖.By construction, 𝑌∈𝑐^𝑎2 and 𝑌 has infinitely many common elements with each 𝑋^𝑘, 𝑘∈ℕ; that is, the selection principle 𝛼₂(𝑐^𝑎2,𝑐^𝑎2) 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 𝑋^𝑘, 𝑘∈ℕ, but on the same (corresponding) positions.
Let for each 𝑘∈ℕ, 𝑥^𝑘 denote the sequence (𝑥^{𝑘𝑚,𝑚})_𝑚∈ℕ. Then each 𝑥^𝑘 converges to 𝑎, so that we have the sequence (𝑥^𝑘∶𝑘∈ℕ) of sequences converging to 𝑎. Let 2=𝑝₁<𝑝₂<𝑝₃<⋯ be a sequence of prime natural numbers. Take sequence 𝑥¹=(𝑥^1𝑚,𝑚)_𝑚∈ℕ. For each 𝑖∈ℕ, replace the elements of 𝑥¹ on the positions 𝑝^ℎ𝑖, ℎ∈ℕ, by the corresponding elements of the sequence 𝑥^𝑖+1. One obtains the sequence (𝑧_𝑚)_𝑚∈ℕ converging to 𝑎 which has infinitely many common elements with each 𝑥^𝑘 on the same positions as in 𝑥^𝑘. Define now the double sequence 𝑌=(𝑦_𝑠,𝑡)_{𝑠,𝑡∈ℕ} so that 𝑦_𝑠,𝑠=𝑧_𝑠, 𝑠∈ℕ, and 𝑦_𝑠,𝑡=𝑎 whenever 𝑠≠𝑡. By construction, 𝑌∈𝑐^𝑎2 and has infinitely many common positions with each 𝑋^𝑘.

The following definition gives a double sequence version of the selection property 𝛼₂(𝒜,ℬ).

Definition 2.7. Let 𝒜 and ℬ be subclasses of 𝑐^𝑎2. Then 𝛼^2(𝑑)(𝒜,ℬ) denotes the selection hypothesis: for each double sequence (𝐴_𝑚,𝑛∶𝑚,𝑛∈ℕ) of elements of 𝒜 there is an element B in ℬ such that 𝐵∩𝐴_𝑚,𝑛 is infinite for all (𝑚,𝑛)∈ℕ×ℕ.

Theorem 2.8. Let 𝑎∈ℝ be given. The selection principle 𝛼^2(𝑑)(𝑐^𝑎2,𝑐^𝑎2) is true.

Proof. Let (𝑋^𝑗,𝑘∶𝑗,𝑘∈ℕ) be a double sequence of elements in 𝑐^𝑎2 and let 𝑋^𝑗,𝑘=(𝑥^{𝑗,𝑘𝑚,𝑛})_{𝑚,𝑛∈ℕ}. In a standard way (see [2]) form from this double sequence a sequence (𝑋^𝑖∶𝑖∈ℕ) of double sequences 𝑋^𝑖=(𝑥^{𝑖𝑚,𝑛})_{𝑚,𝑛∈ℕ}. Apply now Lemma 2.5 to this sequence and find a double sequence 𝑌∈𝑐^𝑎2 such that 𝑌∩𝑋^𝑖 is infinite for each 𝑖∈ℕ. But then 𝑌∩𝑋^𝑗,𝑘 is infinite for all 𝑗,𝑘∈ℕ.

Remark 2.9. Notice that the double sequence 𝑌 from the proofs of Lemma 2.5 and Theorem 2.8 satisfies: (a) 𝑌 is bounded; (b) 𝑌 is regular, and lim_𝑛→∞𝑦_𝑚,𝑛=lim_𝑚→∞𝑦_𝑚,𝑛=𝑎 for each 𝑚∈ℕ and each 𝑛∈ℕ.

Theorem 2.10. Let 𝑎∈ℝ and let (𝑋^𝑘∶𝑘∈ℕ) be a sequence of double sequences in 𝑐^𝑎2, 𝑋^𝑘=(𝑥^{𝑘𝑚,𝑛})_{𝑚,𝑛∈ℕ}. Then there is a double sequence 𝑌=(𝑦_𝑠,𝑡)_{𝑠,𝑡∈ℕ} in 𝑐^𝑎2 such that for each 𝑘∈ℕ the set {(𝑠,𝑡)∈ℕ×ℕ∶𝑦_𝑠,𝑡=𝑥^{𝑘𝑚,𝑛}forsome(𝑚,𝑛)∈ℕ×ℕ} is infinite.

Proof. The double sequence 𝑌 is defined in the following way.
Let 𝑘∈ℕ. There is 𝑖_𝑘∈ℕ such that |𝑥^{𝑘𝑚,𝑛}−𝑎|<2^−𝑘 for all 𝑚,𝑛≥𝑖_𝑘. Let 𝑠^∗=𝑖^𝑘𝑖,for𝑠=𝑘,_𝑘𝑡+𝑝,for𝑠=𝑘+𝑝,𝑝∈ℕ,_∗=𝑖^𝑘𝑖,for𝑡=𝑘,_𝑘+𝑝,for𝑡=𝑘+𝑝,𝑝∈ℕ.(2.3) For 𝑡≥𝑘 let 𝑦_𝑘,𝑡=𝑥^𝑘𝑖_𝑘,𝑡∗, and for 𝑠≥𝑘 let 𝑦_𝑠,𝑘=𝑥^𝑘𝑠_∗,𝑖𝑘. The double sequence 𝑌=(𝑦_𝑠,𝑡)_{𝑠,𝑡∈ℕ} constructed in this way is as required, because 𝑌 has the following properties:(1)𝑌∈𝑐^𝑎2;(2)the set 𝐵^𝑘={𝑦_𝑘,𝑡∶𝑡≥𝑘}∪{𝑦_𝑠,𝑘∶𝑠≥𝑘} is a subset of 𝐴^𝑘={𝑥^{𝑘𝑚,𝑛}∶𝑚,𝑛∈ℕ};(3)for each 𝑘∈ℕ, 𝐵^𝑘 is countable;(4)⋃_𝑘∈ℕ𝐵^𝑘={𝑦_𝑠,𝑡∶𝑠,𝑡∈ℕ}.

Another similar result is given in the next theorem.

Theorem 2.11. Let 𝑎∈ℝ and let (𝑋^𝑘∶𝑘∈ℕ) be a sequence of double sequences in 𝑐^𝑎2, 𝑋^𝑘=(𝑥^{𝑘𝑚,𝑛})_{𝑚,𝑛∈ℕ}. Then there is a double sequence 𝑌=(𝑦_𝑠,𝑡)_{𝑠,𝑡∈ℕ} in 𝑐^𝑎2 which has one common row with 𝑋^𝑘 for each 𝑘∈ℕ.

Proof. For each 𝑘∈ℕ there is 𝑛₀(𝑘)∈ℕ such that |𝑥^{𝑘𝑚,𝑛}−𝑎|<2^−𝑘 for all 𝑚,𝑛≥𝑛₀(𝑘), 𝑛₀(𝑘₁)>𝑛₀(𝑘₂) whenever 𝑘₁>𝑘₂, and 𝑛₀(𝑘)≥min{𝑖(𝑘)∈ℕ∶|𝑥^{𝑘+1𝑚,𝑛}−𝑎|<2^−𝑘forall𝑚,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 𝑋^𝑘, that is 𝑦_{𝑛0(𝑘),𝑛}=𝑥^𝑘𝑛_0(𝑘),𝑛 (𝑛∈ℕ), and 𝑦_𝑠,𝑡=𝑎 otherwise. Let us prove that 𝑌∈𝑐^𝑎2. Indeed, if 𝜀>0 is given, then choose 𝑝∈ℕ such that 2^−𝑝<𝜀. Then for each 𝑘∈ℕ we have |𝑥^{𝑘𝑚,𝑛}−𝑎|<𝜀 for all 𝑚,𝑛≥𝑝. By construction of 𝑌 we have actually that |𝑦_𝑚,𝑛−𝑎|<𝜀 for all 𝑚,𝑛≥𝑝, that is 𝑌∈𝑐^𝑎2.

Consider now an order on the set ℕ×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. Let 𝒜 and ℬ be subclasses of 𝑐^𝑎2. Then 𝖲^𝜑1(𝒜,ℬ) denotes the selection hypothesis: for each sequence (𝐴_𝑛∶𝑛∈ℕ) of elements of 𝒜 there is an element 𝐵=(𝑏_{𝜑−1(𝑛)})_𝑛∈ℕ in ℬ such that 𝑏_{𝜑−1(𝑛)}∈𝐴_𝑛 for all 𝑛∈ℕ.

Theorem 2.13. Let 𝑎∈ℝ and let ≤_𝜑 be as previously mentioned. Then the selection hypothesis 𝖲^𝜑1(𝑐^𝑎2,𝑐^𝑎2) is satisfied.

Proof. Let (𝑋^𝑘∶𝑘∈ℕ), 𝑋^𝑘=(𝑥^{𝑘𝑚,𝑛})_{𝑚,𝑛∈ℕ}, be a sequence in 𝑐^𝑎2. Construct a double sequence 𝑌=(𝑦_𝑠,𝑡)_{𝑠,𝑡∈ℕ} as follows.
Fix 𝑘∈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 |𝑥^{𝑘𝑚,𝑛}−𝑎|<2^{−𝑝(𝑘)} for all 𝑚,𝑛≥𝑛₀(𝑘). Set 𝑦_{𝑠(𝑘),𝑡(𝑘)}=𝑥^𝑘𝑛_{0(𝑘),𝑛0(𝑘)} and 𝑌=(𝑦_{𝑠(𝑘),𝑡(𝑘)})_𝑘∈ℕ. Then, by the construction, 𝑌∈𝑐^𝑎2 and 𝑌 have exactly one common element with 𝑋^𝑘 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 the 𝖯-limits.

Acknowledgments

The authors are supported by MES RS.