Abstract and Applied Analysis

Volume 2011, Article ID 213816, 8 pages

http://dx.doi.org/10.1155/2011/213816

## Summability of Sequences and Selection Properties

^{1}Technical Faculty Čačak, Svetog Save 65, 32000 Čačak, Serbia^{2}Faculty of Sciences and Mathematics, University of Niš, 18000 Niš, Serbia^{3}Singidunum University, Belgrade, Serbia

Received 9 February 2011; Accepted 10 April 2011

Academic Editor: Jean Pierre Gossez

Copyright © 2011 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 prove that some classes of summable sequences of positive real numbers satisfy several selection principles related to a special kind of convergence.

#### 1. Introduction

By , , and we denote the set of natural numbers, real numbers, and the extended real line , respectively.

Let denote the set of sequences of positive real numbers.

We begin with the following definitions of selection principles.

Let and be nonempty subsets of . Then the symbol denotes the selection principle.

For each sequence of elements of there is a sequence such that for each .

The following infinitely long game is naturally associated to the previous selection principle. Two players, ONE and TWO, play a round for each positive integer. In the -th round ONE chooses a sequence , and TWO responds by choosing an element . TWO wins a play if ; otherwise, ONE wins.

Another selection principle is defined as follows.For each sequence of elements of there is a sequence such that is finite for each .

It is clear how the corresponding game is defined.

A strategy of a player is a function from the set of all finite sequences of moves of the other player into the set of admissible moves of the strategy owner.

A strategy 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 ; , .

In this paper we introduce also the following game. Let be a fixed (but arbitrary) natural number. We define the game for two players, ONE and TWO, who play a round for each . In the -th round ONE plays a sequence , and TWO responds by choosing a finite set . In the -th round, , ONE plays a sequence , and TWO responds by choosing an element . TWO wins a play if the sequence belongs to ; 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 subclasses and of satisfy certain selection principles and game theoretical statements (for and classes of divergent sequences related to celebrated Karamata's theory of regular variation [3–6] see [7–12], and for and 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:

Notice that the sequence , , belongs to the class , so that all the classes above are nonempty.

Theorem 2.1. *For each and each TWO has a winning coding strategy in the game .*

*Proof. *Let denote a strategy of TWO, and let and be fixed. Suppose that in the first round ONE chooses a sequence from . There is such that , and thus . Player TWO plays —a finite subset of .

In the second round ONE chooses a sequence , and then TWO responds by choosing such that (which is possible because ).

In the -th round, , ONE chooses , and TWO's response is such that , and so on.

Set for and for . Let us prove . We have
On the other hand,
That is, .

Corollary 2.2. *For each and each the selection principle is true.*

Notice that one can prove a refinement of Theorem 2.1 (and Corollary 2.2) in the sense that it is possible to have additional control of selections giving the sequence . For this we need the following definitions and notation.

*Definition 2.3 (see [13]). * A sequence is said to belong to the class if for each it satisfies
where denotes the integer part of .

*Definition 2.4 (see [9]). * For a sequence , the *Landau-Hurwicz sequence * of is defined by

Given a sequence we denote by the sequence defined by

Let be the set of all sequences such that .

Theorem 2.5. *For each and each TWO has a winning coding strategy in the game .*

*Proof. *The strategy of player TWO and the sequence are actually from the proof of Theorem 2.1. Therefore, . Besides, since, by construction, the series
is convergent, we have

Consider now the sequence . This sequence is convergent (by the d'Alembert criterion), and let be its limit. It remains to prove . It is enough to prove

First, notice that
Thus we get
That is (2.9), since by (2.8) and the fact that for sufficiently large it holds
we have
The theorem is proved.

Corollary 2.6. *The selection principle is true.*

The following two theorems give other selection results for defined classes of sequences: one of the -type and the other of the -type.

Theorem 2.7. *For each the selection principle is satisfied.*

*Proof. *Consider first the case . Let , , be a sequence of elements of . For each let , , be a finite subset of such that . Arrange now , , , in the sequence in which the position of an element is determined first by and then by , that is,
We have
where is the last element of belonging to the sequence . Therefore,
That is, .

Suppose now that . This case is treated similarly to the previous case, but here we require for each ; the sequence is formed in a similar way as in the first case. So we have for each , hence
That is, . The theorem is proved.

Theorem 2.8. *For each and each the selection principle is true.*

*Proof. *Let , , be a sequence of elements in . For each take so that (which is possible since as ) and for . Then the sequence witnesses that the statement is true, because
That is, .

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 . In [19] we proposed the use of this technique (and these selection principles) in other fields of mathematics and its applications.

Let, as before, and be certain nonempty subfamilies of . Then the symbol , , denotes the selection hypothesis that for each sequence of elements of there is an element such that:: for each the set is infinite;: for infinitely many the set is infinite;: for infinitely many the set is nonempty.

Theorem 2.9. *For each and each the selection principles , , are satisfied.*

*Proof. *We prove that the principle is true (hence also and ). Let , , be a sequence of sequences from . Let be such that . For let be a natural number such that . Consider the sequence defined in this way:
Then is infinite for each . Further, because

We construct now a new sequence in the way described in Table 1.

Evidently, is infinite for each . Also, , that is, . By a minor modification of the proof of Theorem 2.5 we obtain . This means .

#### Acknowledgments

The authors are supported by the Ministry of Science and Technological Development of the Republic of Serbia. They thank the referees for their several useful comments.

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