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Abstract and Applied Analysis
Volume 2012, Article ID 497594, 6 pages
http://dx.doi.org/10.1155/2012/497594
Research Article

Double Sequences and Selections

1Technical Faculty in Čačak, University of Kragujevac, Svetog Save 65, 32000 Čačak, Serbia
2Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia
3Department of Applied Mathematics, Singidunum University, 11070 Belgrade, Serbia

Received 6 May 2012; Accepted 18 June 2012

Academic Editor: Márcia Federson

Copyright © 2012 Dragan Djurčić et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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=(𝑥𝑚,𝑛)𝑚,𝑛 converges to 𝑎 (notation 𝖯-lim𝑋=𝑎 or 𝖯-lim𝑥𝑚,𝑛=𝑎), if for every 𝜀>0 there is 𝑛0 such that |𝑥𝑚,𝑛𝑎|<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 𝑐𝑎2 the set of all double real sequences converging to a point 𝑎 in Pringsheim's sense.

A considerable number of papers which appeared in recent years study the set 𝑐𝑎2 and its subsets from various points of view (see [413]). 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 (𝑋𝑛𝑛) 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 (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 [1619]). 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 𝑐𝑎2 and will be defined below.

If 𝖯-lim|𝑋|=, (equivalently, for every 𝑀>0 there are 𝑛1,𝑛2 such that |𝑥𝑚,𝑛|>𝑀 whenever 𝑚𝑛1, 𝑛𝑛2), then 𝑋 is said to be definitely divergent.

A double sequence 𝑋=(𝑥𝑚,𝑛)𝑚,𝑛 is bounded if there is 𝑀>0 such that |𝑥𝑚,𝑛|<𝑀 for all 𝑚,𝑛.

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

A number 𝐿 is said to be a Pringsheim limit point of a double sequence 𝑋=(𝑥𝑚,𝑛)𝑚,𝑛 if there exist two increasing sequences 𝑚1<𝑚2<𝑚𝑖, and 𝑛1<𝑛2<𝑛𝑖, such that lim𝑖𝑥𝑚𝑖,𝑛𝑖=𝐿.(1.1)

In [20], Hardy introduced the notion of regular convergence for double sequences: a double sequence 𝑋=(𝑥𝑚,𝑛)𝑚,𝑛 regularly converges to a point 𝑎 if it 𝖯-converges to 𝑎 and for each 𝑚 and each 𝑛 there exist the following two limits: lim𝑛𝑥𝑚,𝑛=𝑅𝑚,lim𝑚𝑥𝑚,𝑛=𝐶𝑛.(1.2)

The symbol 𝑐𝑎2 denotes the set of elements (𝑥𝑚,𝑛)𝑚,𝑛 in 𝑐𝑎2 which are bounded, regular and such that lim𝑚𝑥𝑚,𝑛=lim𝑛𝑥𝑚,𝑛=𝑎.

2. Results

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

Definition 2.1. Let 𝒜 and be subclasses of 𝑐𝑎2. Then 𝖲1(𝖽)(𝒜,) denotes the selection hypothesis: for each double sequence (𝐴𝑚,𝑛𝑚,𝑛) of elements of 𝒜 there are elements 𝑎𝑚,𝑛𝐴𝑚,𝑛 such that the double sequence (𝑎𝑚,𝑛)𝑚,𝑛 belongs to .

Theorem 2.2. For 𝑎 the selection principle 𝖲1(𝖽)(𝑐𝑎2,𝑐𝑎2) is true.

Proof. Let (𝑋𝑗,𝑘𝑗,𝑘) be a double sequence of elements in 𝑐𝑎2. Suppose that 𝑋𝑗,𝑘=(𝑥𝑗,𝑘𝑚,𝑛)𝑚,𝑛 for all 𝑗,𝑘. Let us construct a double sequence 𝑌=(𝑦𝑚,𝑛)𝑚,𝑛 in the following way:(1)𝑦1,1=𝑥𝑚1,11,𝑚1𝑋1,1, where 𝑚1 is such that |𝑥1,1𝑚,𝑛𝑎|1/2 for each 𝑚𝑚1 and each 𝑛𝑚1.(2)𝑦1,2=𝑥𝑚1,22,𝑚2𝑋1,2, where 𝑚2 is such that |𝑥1,2𝑚,𝑛𝑎|1/22 for each 𝑚𝑚2 and each 𝑛𝑚2.(3)𝑦2,1=𝑥𝑚2,13,𝑚3𝑋2,1 with 𝑚3 such that |𝑥2,1𝑚,𝑛𝑎|1/22 for each 𝑚𝑚3 and each 𝑛𝑚3.(4)𝑦2,2=𝑥𝑚2,24,𝑚4𝑋2,2 with 𝑚4 such that |𝑥2,2𝑚,𝑛𝑎|1/22 for each 𝑚𝑚4 and each 𝑛𝑚4.
In general, for 𝑠,𝑡, 𝑞=max{𝑠,𝑡}3, we put 𝑦𝑠,𝑡=𝑥𝑚𝑠,𝑡𝑝,𝑚𝑝, where 𝑝=(𝑞1)2+𝑡,if𝑞=𝑠,(𝑞1)2+2𝑡𝑠,if𝑞=𝑡,(2.1) and |𝑥𝑠,𝑡𝑚,𝑛𝑎|1/2𝑞 for each 𝑚𝑚𝑝 and each 𝑛𝑚𝑝.
We prove that the double sequence 𝑌=(𝑦𝑚,𝑛)𝑚,𝑛𝑐𝑎2. Let 𝜀>0 be given. Pick 𝑟 such that 1/2𝑟<𝜀. For each 𝑚𝑟 and each 𝑛𝑟, by construction of 𝑌, we have |𝑦𝑚,𝑛𝑎|1/2𝑟<𝜀. This just means 𝑌𝑐𝑎2. As 𝑦𝑚,𝑛𝑋𝑚,𝑛 for all 𝑚,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 lim𝑚𝑦𝑚,𝑛=lim𝑛𝑦𝑚,𝑛=𝑎 for each 𝑚 and each 𝑛, that is 𝑌𝑐𝑎2.

Definition 2.4 (see [15]). Let 𝒜 and be subclasses of 𝑐𝑎2. Then 𝛼2(𝒜,) denotes the selection hypothesis: for each sequence (𝐴𝑛𝑛) of elements of 𝒜 there is an element 𝐵 in such that 𝐵𝐴𝑛 is infinite for all 𝑛.

Lemma 2.5. For 𝑎, the selection principle 𝛼2(𝑐𝑎2,𝑐𝑎2) is satisfied.

Proof. Let (𝑆𝑘𝑘) be a sequence of elements from 𝑐𝑎2 and let for each 𝑘, 𝑆𝑘=(𝑥𝑘𝑚,𝑛)𝑚,𝑛.(1)Form first an increasing sequence 𝑗1<𝑗2<<𝑗𝑖< in so that:(a)𝑗1=min{𝑛0|𝑥1𝑚,𝑛𝑎|1/2forall𝑚𝑛0andforall𝑛𝑛0};(b)Let 𝑖2. Find 𝑝𝑖=min{𝑛0|𝑥𝑖𝑚,𝑛𝑎|1/2𝑖forall𝑚,𝑛𝑛0}, and then define 𝑗𝑖=𝑝𝑖,if𝑝𝑖>𝑗𝑖1;𝑗𝑖1+1,if𝑝𝑖𝑗𝑖1.(2.2)(2)Define now a double sequence 𝑌=(𝑦𝑠,𝑡)𝑠,𝑡 in this way:(a)𝑦𝑠,𝑡=𝑥1𝑠,𝑡 for each 1𝑠<𝑗2, 𝑡, and each 1𝑡<𝑗2, 𝑠;(b)for 𝑖2, 𝑦𝑠,𝑡=𝑥𝑖𝑠,𝑡, for 𝑗𝑖𝑠<𝑗𝑖+1, 𝑡𝑗𝑖, and 𝑗𝑖𝑡<𝑗𝑖+1, 𝑠𝑗𝑖.By construction, 𝑌𝑐𝑎2 and 𝑌 has infinitely many common elements with each 𝑋𝑘, 𝑘; that is, the selection principle 𝛼2(𝑐𝑎2,𝑐𝑎2) 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, but on the same (corresponding) positions.
Let for each 𝑘, 𝑥𝑘 denote the sequence (𝑥𝑘𝑚,𝑚)𝑚. Then each 𝑥𝑘 converges to 𝑎, so that we have the sequence (𝑥𝑘𝑘) of sequences converging to 𝑎. Let 2=𝑝1<𝑝2<𝑝3< be a sequence of prime natural numbers. Take sequence 𝑥1=(𝑥1𝑚,𝑚)𝑚. For each 𝑖, replace the elements of 𝑥1 on the positions 𝑝𝑖, , by the corresponding elements of the sequence 𝑥𝑖+1. One obtains the sequence (𝑧𝑚)𝑚 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 so that 𝑦𝑠,𝑠=𝑧𝑠, 𝑠, and 𝑦𝑠,𝑡=𝑎 whenever 𝑠𝑡. By construction, 𝑌𝑐𝑎2 and has infinitely many common positions with each 𝑋𝑘.

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

Definition 2.7. Let 𝒜 and be subclasses of 𝑐𝑎2. Then 𝛼2(𝑑)(𝒜,) denotes the selection hypothesis: for each double sequence (𝐴𝑚,𝑛𝑚,𝑛) of elements of 𝒜 there is an element B in such that 𝐵𝐴𝑚,𝑛 is infinite for all (𝑚,𝑛)×.

Theorem 2.8. Let 𝑎 be given. The selection principle 𝛼2(𝑑)(𝑐𝑎2,𝑐𝑎2) is true.

Proof. Let (𝑋𝑗,𝑘𝑗,𝑘) be a double sequence of elements in 𝑐𝑎2 and let 𝑋𝑗,𝑘=(𝑥𝑗,𝑘𝑚,𝑛)𝑚,𝑛. In a standard way (see [2]) form from this double sequence a sequence (𝑋𝑖𝑖) of double sequences 𝑋𝑖=(𝑥𝑖𝑚,𝑛)𝑚,𝑛. Apply now Lemma 2.5 to this sequence and find a double sequence 𝑌𝑐𝑎2 such that 𝑌𝑋𝑖 is infinite for each 𝑖. But then 𝑌𝑋𝑗,𝑘 is infinite for all 𝑗,𝑘.

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 lim𝑛𝑦𝑚,𝑛=lim𝑚𝑦𝑚,𝑛=𝑎 for each 𝑚 and each 𝑛.

Theorem 2.10. Let 𝑎 and let (𝑋𝑘𝑘) be a sequence of double sequences in 𝑐𝑎2, 𝑋𝑘=(𝑥𝑘𝑚,𝑛)𝑚,𝑛. Then there is a double sequence 𝑌=(𝑦𝑠,𝑡)𝑠,𝑡 in 𝑐𝑎2 such that for each 𝑘 the set {(𝑠,𝑡)×𝑦𝑠,𝑡=𝑥𝑘𝑚,𝑛forsome(𝑚,𝑛)×} is infinite.

Proof. The double sequence 𝑌 is defined in the following way.
Let 𝑘. There is 𝑖𝑘 such that |𝑥𝑘𝑚,𝑛𝑎|<2𝑘 for all 𝑚,𝑛𝑖𝑘. Let 𝑠=𝑖𝑘𝑖,for𝑠=𝑘,𝑘𝑡+𝑝,for𝑠=𝑘+𝑝,𝑝,=𝑖𝑘𝑖,for𝑡=𝑘,𝑘+𝑝,for𝑡=𝑘+𝑝,𝑝.(2.3) For 𝑡𝑘 let 𝑦𝑘,𝑡=𝑥𝑘𝑖𝑘,𝑡, and for 𝑠𝑘 let 𝑦𝑠,𝑘=𝑥𝑘𝑠,𝑖𝑘. The double sequence 𝑌=(𝑦𝑠,𝑡)𝑠,𝑡 constructed in this way is as required, because 𝑌 has the following properties:(1)𝑌𝑐𝑎2;(2)the set 𝐵𝑘={𝑦𝑘,𝑡𝑡𝑘}{𝑦𝑠,𝑘𝑠𝑘} is a subset of 𝐴𝑘={𝑥𝑘𝑚,𝑛𝑚,𝑛};(3)for each 𝑘, 𝐵𝑘 is countable;(4)𝑘𝐵𝑘={𝑦𝑠,𝑡𝑠,𝑡}.

Another similar result is given in the next theorem.

Theorem 2.11. Let 𝑎 and let (𝑋𝑘𝑘) be a sequence of double sequences in 𝑐𝑎2, 𝑋𝑘=(𝑥𝑘𝑚,𝑛)𝑚,𝑛. Then there is a double sequence 𝑌=(𝑦𝑠,𝑡)𝑠,𝑡 in 𝑐𝑎2 which has one common row with 𝑋𝑘 for each 𝑘.

Proof. For each 𝑘 there is 𝑛0(𝑘) such that |𝑥𝑘𝑚,𝑛𝑎|<2𝑘 for all 𝑚,𝑛𝑛0(𝑘), 𝑛0(𝑘1)>𝑛0(𝑘2) whenever 𝑘1>𝑘2, and 𝑛0(𝑘)min{𝑖(𝑘)|𝑥𝑘+1𝑚,𝑛𝑎|<2𝑘forall𝑚,𝑛𝑖(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=𝑥𝑘𝑛0(𝑘),𝑛 (𝑛), and 𝑦𝑠,𝑡=𝑎 otherwise. Let us prove that 𝑌𝑐𝑎2. Indeed, if 𝜀>0 is given, then choose 𝑝 such that 2𝑝<𝜀. Then for each 𝑘 we have |𝑥𝑘𝑚,𝑛𝑎|<𝜀 for all 𝑚,𝑛𝑝. By construction of 𝑌 we have actually that |𝑦𝑚,𝑛𝑎|<𝜀 for all 𝑚,𝑛𝑝, that is 𝑌𝑐𝑎2.

Consider now an order on the set ×. Let 𝜑× be a bijection. Set (𝑚1,𝑛1)𝜑(𝑚2,𝑛2)𝜑(𝑚1,𝑛1)𝜑(𝑚2,𝑛2), where is the natural order in .

Definition 2.12. Let 𝒜 and be subclasses of 𝑐𝑎2. Then 𝖲𝜑1(𝒜,) denotes the selection hypothesis: for each sequence (𝐴𝑛𝑛) of elements of 𝒜 there is an element 𝐵=(𝑏𝜑1(𝑛))𝑛 in such that 𝑏𝜑1(𝑛)𝐴𝑛 for all 𝑛.

Theorem 2.13. Let 𝑎 and let 𝜑 be as previously mentioned. Then the selection hypothesis 𝖲𝜑1(𝑐𝑎2,𝑐𝑎2) is satisfied.

Proof. Let (𝑋𝑘𝑘), 𝑋𝑘=(𝑥𝑘𝑚,𝑛)𝑚,𝑛, be a sequence in 𝑐𝑎2. Construct a double sequence 𝑌=(𝑦𝑠,𝑡)𝑠,𝑡 as follows.
Fix 𝑘. Let (𝑠(𝑘),𝑡(𝑘))=𝜑1(𝑘), and let 𝑝(𝑘)=max{𝑠(𝑘),𝑡(𝑘)}. There is 𝑛0(𝑘) such that |𝑥𝑘𝑚,𝑛𝑎|<2𝑝(𝑘) for all 𝑚,𝑛𝑛0(𝑘). Set 𝑦𝑠(𝑘),𝑡(𝑘)=𝑥𝑘𝑛0(𝑘),𝑛0(𝑘) and 𝑌=(𝑦𝑠(𝑘),𝑡(𝑘))𝑘. Then, by the construction, 𝑌𝑐𝑎2 and 𝑌 have exactly one common element with 𝑋𝑘 for each 𝑘, 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.

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