I-Pre-Cauchy Double Sequences and Orlicz Functions

Let   ij x x  be a double sequence and let M be a bounded Orlicz function. We prove that x is I-pre-Cauchy if and only if 2 2 , , 1 lim mn 0. ij pq i p m j q n x x I M m n                 This implies a theorem due to Connor, Fridy and Klin [1], and Vakeel A. Khan and Q. M. Danish Lohani [2].

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Introduction
The concept of statistical convergence was first defined by Steinhaus [3] at a conference held at Wroclaw University, Poland in 1949 and also independently by Fast [4], Buck [5] and Schoenberg [6] for real and complex sequences.Further this concept was studied by Salat [7], Fridy [8], Connor [9] and many others.Statistical convergence is a generalization of the usual notation of convergence that parallels the usual theory of convergence.A sequence  is said to be statistically convergent to if for a given L 0 Connor, Fridy and Klin [1] proved that statistically convergent sequences are statistically pre-cauchy and any bounded statistically pre-cauchy sequence with a nowhere dense set of limit points is statistically convergent.They also gave an example showing statistically pre-cauchy sequences are not necessarily statistically convergent (see [10]).
Throughout a double sequence is denoted by A double sequence is a double infinite array of elements ij x   for all , .i j   The initial works on double sequences is found in Bromwich [11], Tripathy [12], Basarir and Solancan [13] and many others.
where the vertical bars indicate the number of elements in the set.
is called statistically pre-cauchy if for every 0 If convexity of M is replaced by x  , then it is called a Modulus function (see Maddox [14]).An Orlicz function may be bounded or unbounded.For example, is bounded (see Maddox [14]).
Lindenstrauss and Tzafriri [15] used the idea of Orlicz functions to construct the sequence space, 1 : , The space M  is a Banach space with the norm 1 inf 0 : The space M  is closely related to the space p  which is an Orlicz sequence space with An Orlicz function M is said to satisfy 2 condition for all values of  x if there exists a constant such that The study of Orlicz sequence spaces have been made recently by various authors [1,2,[16][17][18][19][20]).
In [1], Connor,Fridy and Klin proved that a bounded sequence The notion of I-convergence is a generalization of statistical convergence.At the initial stage it was studied by Kostyrko, Salat, Wilezynski [21].Later on it was studied by Salat, Tripathy, Ziman [22] and Demirci [23], Tripathy and Hazarika [24][25][26].Here we give some preliminaries about the notion of I-convergence.
Definition 1.4.[20,27] Let X be a non empty set.Then a family of sets ( denoting the power set of X) is said to be an ideal in X if A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal J I  containing I as a subset.
For each ideal I, there is a filter   £ I corresponding to I. i.e.
In this case we write lim . is said to be filter on X if and only if (i)

Main Results
In this article we establish the criterion for any arbitrary double sequence to be I-pre-cauchy.

A x x m n IN M i p m j q n mn
Now by ( 1) and ( 2) we have Now conversely suppose that x is I-pre-Cauchy, and that  has been given.
Then we have where, .
x  be a double sequence and let M be a bounded Orlicz function then x is I-convergent to L if and only if 1 lim 0, for some 0.

M x x 
We can prove this in the similar manner as in the proof of Corollary 2.3.

( 3 )
Since x is I-pre-Cauchy, there is an IN such that the right hand side of (3) is less than  for all .Hence ,