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 converges to (notation or ), if for every there is such that for all (see , and also [2, 3]). The limit is called the Pringsheim limit of .

In this paper we denote by the set of all double real sequences converging to a point in Pringsheim's sense.

A considerable number of papers which appeared in recent years study the set and its subsets from various points of view (see ). 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 of double sequences that belong to one class we select from each a subset by a prescribed procedure so that 's may be arranged to a new double sequence which belongs to another class (not necessarily distinct from ) of double sequences. (For selection principles theory see [14, 15], and for selection properties of some classes of simple sequences see ). Moreover, our investigation suggests also introduction of new selection principles: instead of a sequence of double sequences from a class we start with a double sequence of double sequences from (see Definitions 2.1 and 2.7). The classes of double sequences considered in this article are subsets of the class and will be defined below.

If , (equivalently, for every there are such that whenever , ), then is said to be definitely divergent.

A double sequence is bounded if there is such that for all .

Notice that a convergent double sequence need not be bounded.

A number is said to be a Pringsheim limit point of a double sequence if there exist two increasing sequences and such that

In , Hardy introduced the notion of regular convergence for double sequences: a double sequence regularly converges to a point if it converges to and for each and each there exist the following two limits:

The symbol denotes the set of elements in which are bounded, regular and such that .

2. Results

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

Definition 2.1. Let and be subclasses of . Then denotes the selection hypothesis: for each double sequence of elements of there are elements such that the double sequence belongs to .

Theorem 2.2. For the selection principle is true.

Proof. Let be a double sequence of elements in . Suppose that for all . Let us construct a double sequence in the following way:(1), where is such that for each and each .(2), where is such that for each and each .(3) with such that for each and each .(4) with such that for each and each .
In general, for , , we put , where and for each and each .
We prove that the double sequence . Let be given. Pick such that . For each and each , by construction of , we have . This just means . As for all , the theorem is proved.

Remark 2.3. The double sequence from the proof of Theorem 2.2 has also the following properties: (i) is bounded; (ii) is regular and for each and each , that is .

Definition 2.4 (see ). Let and be subclasses of . Then denotes the selection hypothesis: for each sequence of elements of there is an element in such that is infinite for all .

Lemma 2.5. For , the selection principle is satisfied.

Proof. Let be a sequence of elements from and let for each , .(1)Form first an increasing sequence in so that:(a);(b)Let . Find , and then define (2)Define now a double sequence in this way:(a) for each , , and each , ;(b)for , , for , , and , .By construction, and has infinitely many common elements with each , ; that is, the selection principle is satisfied.

Remark 2.6. Using the technique from  we can prove that the double sequence in the proof of the previous lemma can be chosen in such a way that has infinitely many common elements with each , , but on the same (corresponding) positions.
Let for each , denote the sequence . Then each converges to , so that we have the sequence of sequences converging to . Let be a sequence of prime natural numbers. Take sequence . For each , replace the elements of on the positions , , by the corresponding elements of the sequence . One obtains the sequence converging to which has infinitely many common elements with each on the same positions as in . Define now the double sequence so that , , and whenever . By construction, and has infinitely many common positions with each .

The following definition gives a double sequence version of the selection property .

Definition 2.7. Let and be subclasses of . Then denotes the selection hypothesis: for each double sequence of elements of there is an element in such that is infinite for all .

Theorem 2.8. Let be given. The selection principle is true.

Proof. Let be a double sequence of elements in and let . In a standard way (see ) form from this double sequence a sequence of double sequences . Apply now Lemma 2.5 to this sequence and find a double sequence such that is infinite for each . But then is infinite for all .

Remark 2.9. Notice that the double sequence from the proofs of Lemma 2.5 and Theorem 2.8 satisfies: (a) is bounded; (b) is regular, and for each and each .

Theorem 2.10. Let and let be a sequence of double sequences in , . Then there is a double sequence in such that for each the set is infinite.

Proof. The double sequence is defined in the following way.
Let . There is such that for all . Let For let , and for let . The double sequence constructed in this way is as required, because has the following properties:(1);(2)the set is a subset of ;(3)for each , is countable;(4).

Another similar result is given in the next theorem.

Theorem 2.11. Let and let be a sequence of double sequences in , . Then there is a double sequence in which has one common row with for each .

Proof. For each there is such that for all , whenever , and . Then the desired double sequence is defined in such a way that its th row is the th row of , that is (), and otherwise. Let us prove that . Indeed, if is given, then choose such that . Then for each we have for all . By construction of we have actually that for all , that is .

Consider now an order on the set . Let be a bijection. Set , where is the natural order in .

Definition 2.12. Let and be subclasses of . Then denotes the selection hypothesis: for each sequence of elements of there is an element in such that for all .

Theorem 2.13. Let and let be as previously mentioned. Then the selection hypothesis is satisfied.

Proof. Let , , be a sequence in . Construct a double sequence as follows.
Fix . Let , and let . There is such that for all . Set and . Then, by the construction, and have exactly one common element with for each , that is 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 limits.

Acknowledgments

The authors are supported by MES RS.

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