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Real Analysis Exchange

Vol. 30(1), 2004/2005, pp. 123-128

Katarzyna Dems, Institute of Mathematics, ?L´od´z Technical University, al. Politechniki 11, 90-924 ?L´od´z, Poland. email:kasidems@p.lodz.pl

ONI-CAUCHY SEQUENCES

Abstract

We studyI-convergence andI-Cauchy sequences in metric spaces whereI?P(Nk) is an ideal containing all singletons andk? {1,2}.

1 Introduction.

Throughout the paper,Ndenotes the set of positive integers,P(X) stands for the power set ofX.For a subsetEof a metric space, clEwill denote the closure ofE.The ball with centerxand radiusrwill be written asB(x, r). Recall that a sequence{xn}n?Nof points in a metric space (X, ρ) is said to bestatistically convergenttox?Xifd(A(ε)) = 0 for eachε >0 where is thedensityof a setE?Nprovided that the limit exists. Several papers on statistical convergence have been published. See [2], [3], [5]. In [4] and [7] an interesting generalization of this notion was proposed. Namely, it is easy to check that the familyId={A?N:d(A) = 0}forms an ideal of subsets ofN.Thus, one may consider an arbitrary idealI?P(N) (assumed non-trivial, i.e.∅ ?=I?=P(N)) to modify the definition of statistical convergence as follows. A sequence{xn}n?Nin (X, ρ) is calledI-convergent tox?X(in shortx=I-limn→∞xn) ifA(ε)?Ifor eachε >0.The article [4] contains many examples and properties ofI-convergence. We shall continue these studies. Our main aim is to prove that, in a complete space (X, ρ), a Cauchy-type condition (borrowed from [3]) is necessary and sufficient for the I-convergence of a given sequence. We also give equivalent formulations of I-Cauchy condition and obtainI-Cauchy condition for double sequences and show some applications.Key Words: Statistical convergence, ideals of sets, Cauchy sequences Mathematical Reviews subject classification: Primary 26A03; Secondary 40A05, 54A20.

Received by the editors December 3, 2003

Communicated by: B. S. Thomson

123

124Katarzyna Dems

Following [4],Iis calledadmissibleif it contains all singletons. The ideal I finof all finite subsets ofNis the smallest admissible ideal inP(N).Ob- serve that the usual convergence, in a given space (X, ρ) coincides withIfin- convergence, and that the usual convergence impliesI-convergence, for any admissible idealI.

2I-Cauchy Condition.

Let (X, ρ) be a metric space andI?P(N) be an admissible ideal. It is easy to check that the classical Cauchy condition for a sequence{xn}n?Nin (X, ρ) is equivalent to the following: for eachε >0 there exists a positive integer ksuch thatρ(xn, xk)< εfor alln≥k.A similar idea was used by Fridy [3] in formulation of the statistical Cauchy condition for a sequence of real numbers. We can modify it to define a Cauchy-type condition associated with I-convergence in (X, ρ). Namely,I-Cauchy conditionreads as follows: for each ε >0 there exists ak?Nsuch that{n?N:ρ(xn, xk)≥ε} ?I.Note that, for I fin,this yields the usual Cauchy condition. Fridy [3] proved that statistical Cauchy condition is equivalent to the statistical convergence of a sequence of reals. However, in any metric space we have the following proposition. Proposition 1.If a sequence of points inXisI-convergent inXthen it fulfillsI-Cauchy condition. Proof.LetI-limn→∞xn=xandε >0.ThusA(ε/2) ={n?N:ρ(xn, x)≥ ε/2} ?I.Pick ank?Nsuch thatk /?A(ε/2).Hence{n?N:ρ(xn, xk)≥

ε} ? {n?N:ρ(xn, x)≥ε/2 orρ(x, xk)≥ε/2}=A(ε/2)?I.In the next theorem we shall show that the equivalence ofI-convergence

andI-Cauchy condition is true for complete metric spaces. Moreover, we shall give a sufficient condition for a metric space to be complete, by the use of I-convergence ofI-Cauchy sequences. The proofs of Proposition 1 and of part (1) in Theorem 2 mimic the arguments of Fridy [3]. Theorem 2.(1). If(X, ρ)is a complete space then everyI-Cauchy sequence inXisI-convergent inX. (2). If everyI-Cauchy sequence inXisI-convergent inXthenXis complete. Proof.(1). Let{xn}n?Nbe anI-Cauchy sequence in a complete space (X, ρ). Considerεm= 1/2m, m?N,and, according toI-Cauchy condition, pick numbersk(m)?N, m?N,such thatAm={n?N:ρ(xn, xk(m))≥ m/2} ?Ifor allm?N.Define inductivelyB1= clB(xk(1), ε1), Bm+1= B m∩clB(xk(m+1), εm+1), m?N.Let us prove thatBm?=∅for eachm?N.

OnI-Cauchy Sequences125

Indeed, we haveA1?Iandxn?B1for alln /?A1.Assume thatm?N andC?Iis a set such thatxn?Bmfor eachn /?C.We haveAm+1?I andxn?clB(xk(m+1),εm+1) for eachn /?Am+1.ThusC?Am+1?Iand x n?Bm+1for alln /?C?Am+1.Since additionallyBm+1?Bmfor all m?N,and the diameter ofBmtends to 0, there is anx?Xsuch that? m?NB m={x},by the Cantor theorem for complete spaces. It suffices to show thatI-limn→∞xn=x.Letε >0 and pick anm?Nsuch thatεm< ε/2.

We have

A(ε)? {n?N:ρ(xn, xk(m)) +ρ(xk(m), x)≥ε}. A(ε)? {n?N:ρ(xn, xk(m)) +ε/2≥ε}={n?N:ρ(xn, xk(m))≥ε/2} ? {n?N:ρ(xn, xk(m))> εm} ?Am?I. (2). Let{xn}n?Nbe a Cauchy sequence in (X, ρ). SinceIis admissible, {xn}n?Nis anI-Cauchy sequence. Thus, by assumption, we haveI-limn→∞xn= xfor somex?X.Putk0= 0 and forε= 1/n,n?N,pick inductively k n?N\({0,...,kn-1} ?A(εn)).Thusρ(xkn, x)<1/nfor everynwhich

implies that limn→∞xkn=x.Consequently, limn→∞xn=x.Note thatI-Cauchy sequences lack some natural properties of Cauchy se-

quences. For instance, a subsequence of anI-Cauchy sequence can be not I-Cauchy which is shown in the following example inspired by [4, Prop. 3.1(ii)]. Example 3.Assume that a metric space (X, ρ) contains at least two distinct pointsxandy.LetI?P(N) be an admissible ideal such that there exists a partition ofNinto pairwise disjoint infinite sets such thatA?IandB /?

I, C /?I.LetA={mn:n?N},B?C={kn:n?N}withmnandkn

strictly increasing. Define{xn}n?Nas follows. Putxkn=xfor alln?N.Let x mn=?xifn?A?B yifn?C. Observe thatI-limn→∞xn=x, thus{xn}n?NisI-Cauchy, by Theorem 2. However, the subsequence{xmn}n?Nis notI-Cauchy (considerε=ρ(x, y).) The statements of Proposition 1 and Theorem 2 (1) were mentioned in [7]. The authors of [7] use however filters rather than ideals. TheirI-Cauchy condition is formulated in a different but equivalent form. Now, we shall prove this equivalence and we add one more equivalent condition. Forε >0 and a sequence{xn}n?Nof points in (X, ρ), we denoteEk(ε) = {n?N:ρ(xn, xk)≥ε}, k?N.

126Katarzyna Dems

Proposition 4.For a sequence{xn}n?Nof points in(X, ρ), the following conditions are equivalent: (1).{xn}n?Nis anI-Cauchy sequence, (2).(cf. [7]) (?ε >0)(?D?I)(?m,n /?D)ρ(xm, xn)< ε, (3).(?ε >0){k?N:Ek(ε)/?I} ?I. Proof.(1)?(2). Letε >0.PutD=Ek(ε/2) wherek?Nis chosen for ε/2 in theI-Cauchy condition for{xn}n?N. ThusD?Iand for anym,n /?D we haveρ(xn, xk)< ε/2 andρ(xm, xk)< ε/2.Henceρ(xn, xm)< εby the triangle inequality. (2)?(3). Letε >0 and letDbe chosen as in (2). We shall show that {k?N:Ek(ε)/?I} ?D. Letk?Nbe such thatEk(ε)/?I.Suppose that k /?D.Pick ann?Ek(ε)\D.Thusρ(xn, xk)≥εby the definition ofEk(ε). Butn,k /?Dimpliesρ(xn, xk)< εby (2), contradiction. (3)?(1). ¿From (3) we have (?ε >0){k?N:Ek(ε)?I} ?=∅which yields (1).3 Double Sequences. In [1], the notion ofI-convergence was extended to the case whenIis an ideal of subsets ofN2and one considers a double sequence{xmn}m,n?Nof points in (X, ρ).(The further generalization deals with multi-indexed sequences and with ideals inP(Nk) fork?N.) Namely, we say that{xmn}m,n?NisI- convergenttox?X(in shortI-limxmn=x) if{(m,n)?N2:ρ(xmn, x)≥ ε} ?Ifor eachε >0.Again an idealI?P(N2) is called admissible if it is non-trivial and contains all singletons. Proposition 5.Let{xmn}m,n?Nbe a sequence of points in a complete metric space(X, ρ)and letI?P(N2)be an admissible ideal. The following conditions are equivalent: (1).{xmn}m,n?Nis anI-convergent sequence, (2).(?ε >0)(?(M,N)?N2){(m,n)?N2:ρ(xmn, xMN)≥ε} ?I. If moreover,Icontains all sets of the form{n} ×N,N× {n}(forn?N), each of the above conditions is equivalent to: (3).(?ε >0)(?l?N)(?M,N≥l){(m,n)?N2:ρ(xmn, xMN)≥ε} ?I.

OnI-Cauchy Sequences127

Proof.To show (1)?(2), fix a bijection?:N2→Nand putJ={?(A) : A?I}. For anx?Xwe haveI-limxmn=x?J-limk→∞x?-1(k)=x. By Proposition 1 and Theorem 2 part (1) this last condition is equivalent to (?ε >0)(?j?N){k?N:ρ(x?-1(k), x?-1(j))≥ε} ?Jwhich in turn is equivalent to (2) when we put (M,N) =?-1(j).Now, assume thatI contains all sets of the form{n} ×N,N× {n}(forn?N). It is obvi- ous that (3)?(2). Let us show implication (1)?(3). Assume that I-limxmn=xand fixε >0 andl?N.Thus{(m,n)?N2:ρ(xmn, x)≥ ε/2} ?I.Since (N× {1,...,l-1})?({1,...,l-1} ×N)?I,we can pick (M,N)? {l,l+ 1,...} × {l,l+ 1,...}withρ(xMN, x)< ε/2.Now, we have{(m,n)?N2:ρ(xmn, xMN)≥ε} ? {(m,n)?N2:ρ(xmn, x)≥ ε/2 orρ(xMN, x)≥ε/2}={(m,n)?N2:ρ(xmn, x)≥ε/2} ?Ias de- sired.Remark.I-Cauchy condition in the form (3) was proved by M´oricz [6] in the particular case whenIconsists of all setsS?N2with two-dimensional den- sityd2(S) equal to 0 whered2(S) = limm,n→∞(1/(mn))·card(S∩({1,...,m} × {1,...,n})) and limm,n→∞zmn=zis meant in the Pringsheim"s sense, that is (?ε >0)(?l?N)(?m,n≥l)|zmn-z|< ε.It is easy to check that the ideal Idefined in such a manner contains all sets of the form{n}×N,N×{n}(for n?N). Acknowledgements.I would like to thank Professor Marek Balcerzak, whose helpful suggestions led to a substantial improvement in this paper. Added in Proof.Recently another proof of Theorem 2(1) has been published in the paper B. K. Lahiri, Pratulananda Das,Further results on I-limit superior and I-limit inferior, Mathematical Communications,8(2003), 151-156.

References

[1] M. Balcerzak, K. Dems,Some types of convergence and related Baire sys- tems, Real Analysis Exchange,31(2004/05), this issue. [2] H. Fast,Sur la convergence statistique, Colloq. Math.,2(1951), 241-244. [3] J. A. Fridy,On statistical convergence, Analysis,5(1985), 301-313. [4] P. Kostyrko, T. Sal´at, W. Wilczy´nski,I-convergence, Real Anal. Exchange,

26(2000/2001), 669-689.

128Katarzyna Dems

[5] H. I. Miller, C. Orhan,On almost convergent and statistically convergent subsequences, Acta Math. Hung.,93(2001), 135-151. [6] F. M´oricz,Statistical convergence of multiple sequences, Arch. Math.,81 (2003), 82-89. [7] F. Nurray, W. H. Ruckle,Generalized statistical convergence and conver- gence free spaces, J. Math. Anal. Appl.,245(2000), 513-527.quotesdbs_dbs19.pdfusesText_25