Research Article (I)-Asymptotically Statistical Equivalence of Sequences of Sets

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Volume 2013, Article ID 602963,6pages http://dx.doi.org/10.1155/2013/602963

Research Article

On

𝑆

𝐿

_𝜆

(𝐼)-Asymptotically Statistical Equivalence of

Sequences of Sets

Ömer K

JGJ

1

and Fat

Jh Nuray

2

1_{Mathematics Education Department, Faculty of Education, Cumhuriyet University, Sıvas, Turkey}

2_{Department of Mathematics, Faculty of Science and Literature, Afyon Kocatepe University, 03200 Afyonkarahısar, Turkey}

Correspondence should be addressed to Fatıh Nuray; [email protected] Received 10 June 2013; Accepted 13 August 2013

Academic Editors: R. Avery and G. Schimperna

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents the notion of𝑆𝐿_𝜆(𝐼)-asymptotically statistical equivalence, which is a natural combination of asymptotic

𝐼-equivalence, and𝜆-statistical equivalence for sequences of sets. We find its relations to 𝐼-asymptotically statistical convergence,

strong𝜆_𝐼-asymptotically equivalence, and strong Cesaro𝐼-asymptotically equivalence for sequences of sets.

1. Introduction and Background

Let 𝜆 = (𝜆_𝑛) be a nondecreasing sequence of positive

numbers tending to∞, such that 𝜆_𝑛+1 ≤ 𝜆_𝑛 + 1, 𝜆₁ = 1.

The generalized de la Vallee-Poussin mean is defined by

𝑡_𝑛(𝑥) = _𝜆1

𝑛𝑘∈𝐼𝑛∑

𝑥_𝑘, ₍₁₎

where𝐼_𝑛= [𝑛 − 𝜆_𝑛+ 1, 𝑛].

A sequence𝑥 = (𝑥_𝑘) is said to be (𝑉, 𝜆)-summable to a

number𝐿 if

lim

𝑛 → ∞𝑡𝑛(𝑥) = 𝐿. (2)

If 𝜆_𝑛 = 𝑛, then (𝑉, 𝜆)-summability reduces to (𝐶,

1)-summability. We write [𝐶, 1] = {𝑥 = (𝑥𝑛) : ∃𝐿 ∈ R, lim_{𝑛 → ∞}1_𝑛 𝑛 ∑ 𝑘=1󵄨󵄨󵄨󵄨𝑥𝑘− 𝐿󵄨󵄨󵄨󵄨 = 0} , [𝑉, 𝜆] ={_{ { 𝑥 = (𝑥_𝑛) : ∃𝐿 ∈ R, lim_{𝑛 → ∞} 1 𝜆_𝑛_{𝑘∈𝐼𝑛}∑󵄨󵄨󵄨󵄨𝑥𝑘− 𝐿󵄨󵄨󵄨󵄨 = 0 } } } , (3)

for the sets of sequences𝑥 = (𝑥_𝑘), which are strongly Cesaro

summable and strongly(𝑉, 𝜆)-summable to 𝐿, that is, 𝑥_𝑘 →

𝐿[𝐶, 1] and 𝑥_𝑘 → 𝐿[𝑉, 𝜆], respectively. Let Λ denote the set

of all nondecreasing sequences𝜆 = (𝜆_𝑛) of positive numbers

tending to∞, such that 𝜆_𝑛+1≤ 𝜆_𝑛and𝜆₁= 1.

Statistical convergence of sequences of points was intro-duced by Fast (see [1]), and under different names, it has been discussed in number theory, trigonometric series, and summability. In 1993, Marouf presented definitions for asymptotically equivalent and asymptotic regular matrices. In 2003, Patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these def-initions and natural regularity conditions for nonnegative

summability matrices. Mursalen defined𝜆-statistical

conver-gence by using the𝜆 sequence. He denoted this new method

by𝑆_𝜆, and found its relation to statistical convergence,[𝐶,

1]-summability, and[𝑉, 𝜆]-summability (see [2]). Savas¸

intro-duced and studied the concepts of strongly 𝜆-summability

and 𝜆-statistical convergence for fuzzy numbers (see [3]).

He also presented asymptotically 𝜆-statistical equivalent

sequences of fuzzy numbers (see [4]). Kostyrko et al. (see [5,

6]) introduced the concept of𝐼-convergence of sequences in a

metric space and studied some properties of this convergence. In addition to these definitions, natural inclusion theorems are also presented. The concept of convergence of sequences of points has been extended by several authors to convergence

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of sequences of sets. One of these extensions that we will consider in this paper is Wijsman convergence. The concept of Wijsman statistical convergence is an implementation of the concept of statistical convergence presented by Nuray and Rhoades (see [7]).

Definition 1. The sequence𝑥 = (𝑥_𝑘) is said to be statistically

convergent to the number𝐿 if for every 𝜀 > 0,

lim 𝑛 → ∞

1

𝑛󵄨󵄨󵄨󵄨{𝑘 ≤ 𝑛 : 󵄨󵄨󵄨󵄨𝑥𝑛− 𝐿󵄨󵄨󵄨󵄨 ≥𝜀}󵄨󵄨󵄨󵄨 = 0. (4)

In this case one writes𝑠𝑡 − lim 𝑥_𝑘= 𝐿 (see [8]).

Definition 2. A family of sets𝐼 ⊆ 2Nis called an ideal if and

only if

(i)0 ∈ 𝐼,

(ii) for each𝐴, 𝐵 ∈ 𝐼 one has 𝐴 ∪ 𝐵 ∈ 𝐼,

(iii) for each𝐴 ∈ 𝐼 and each 𝐵 ⊆ 𝐴 one has 𝐵 ∈ 𝐼 (see [5]).

An ideal is called nontrivial ifN ∉ 𝐼, and nontrivial ideal is

called admissible if{𝑛} ∈ 𝐼 for each 𝑛 ∈ N.

Definition 3. A family of sets𝐹 ⊆ 2Nis a filter inN if and only

if

(i)0 ∉ 𝐼,

(ii) for each𝐴, 𝐵 ∈ 𝐹 one has 𝐴 ∩ 𝐵 ∈ 𝐹,

(iii) for each𝐴 ∈ 𝐹 and each 𝐵 ⊇ 𝐴 one has 𝐵 ∈ 𝐹 (see

[5]).

Proposition 4. 𝐼 is a nontrivial ideal in N if and only if

𝐹 = 𝐹 (𝐼) = {𝑀 = N \ 𝐴 : 𝐴 ∈ 𝐼} , (5)

is a filter inN (see [5]).

Definition 5. Let𝐼 be a nontrivial ideal of subsets of N, and let

(𝑋, 𝑑) be a metric space. A sequence {𝑥_𝑛}_𝑛∈Nof elements of𝑋

is said to be𝐼-convergent to 𝐿. Therefore 𝐿 = 𝐼 − lim_{𝑛 → ∞}𝑥_𝑛

if and only if for each𝜀 > 0 the set

𝐴 (𝜀) = {𝑛 ∈ N : 󵄨󵄨󵄨󵄨𝑥𝑛− 𝐿󵄨󵄨󵄨󵄨 ≥ 𝜀} , (6)

belongs to 𝐼. The number 𝐿 is called the 𝐼 limit of the

sequence𝑥 = (𝑥_𝑛)_𝑛∈N∈ 𝑋 (see [5]).

Definition 6. Let(𝑋, 𝑑) be a metric space. For any non-empty

closed subsets𝐴, 𝐴_𝑘⊆ 𝑋, one says that the sequence {𝐴_𝑘} is

Wijsman convergent to𝐴:

lim

𝑘 → ∞𝑑 (𝑥, 𝐴𝑘) = 𝑑 (𝑥, 𝐴) , (7)

for each𝑥 ∈ 𝑋. In this case one writes 𝑊 − lim_{𝑘 → ∞}𝐴_𝑘 = 𝐴

(see [9,10]).

As an example, consider the following sequence of circles

in the(𝑥, 𝑦)-plane:

𝐴_𝑘 = {(𝑥, 𝑦) : 𝑥2+ 𝑦2+ 2𝑘𝑥 = 0} . (8)

As𝑘 → ∞ the sequence is Wijsman convergent to the 𝑦-axis

𝐴 = {(𝑥, 𝑦) : 𝑥 = 0}.

closed subsets𝐴, 𝐴_𝑘⊆ 𝑋, one says that the sequence {𝐴_𝑘} is

Wijsman statistically convergent to𝐴 if for 𝜀 > 0 and for each

𝑥 ∈ 𝑋, lim 𝑛 → ∞

1

𝑛󵄨󵄨󵄨󵄨{𝑘 ≤ 𝑛 : 󵄨󵄨󵄨󵄨𝑑(𝑥,𝐴𝑘) − 𝑑 (𝑥, 𝐴)󵄨󵄨󵄨󵄨 ≥𝜀}󵄨󵄨󵄨󵄨 = 0. (9)

In this case one writes𝑠𝑡 − lim_𝑊𝐴_𝑘 = 𝐴 or 𝐴_𝑘 → 𝐴(𝑊𝑆)

(see [7]).

𝑊𝑆 := {{𝐴𝑘} : 𝑠𝑡 − lim𝑊𝐴𝑘= 𝐴} , (10)

where𝑊𝑆 denotes the set of Wijsman statistical convergence

sequences.

Also, the concept of bounded sequence for sequences of

sets was given by Nuray and Rhoades (see [7]). Let(𝑋, 𝜌) be a

metric space. For any non-empty closed subsets𝐴_𝑘of𝑋, we

say that the sequence{𝐴_𝑘} is bounded if sup_𝑘𝑑(𝑥, 𝐴_𝑘) < ∞

for each𝑥 ∈ 𝑋.

closed subsets𝐴, 𝐴_𝑘 ⊆ 𝑋, we say that the sequence {𝐴_𝑘}

is Wijsman Cesaro summable to 𝐴 if {𝑑(𝑥, 𝐴_𝑘)} is Cesaro

summable to{𝑑(𝑥, 𝐴)}; that is, for each 𝑥 ∈ 𝑋,

lim 𝑛 → ∞ 1 𝑛 𝑛 ∑ 𝑘=1 𝑑 (𝑥, 𝐴_𝑘) = 𝑑 (𝑥, 𝐴) , (11)

and one says that the sequence {𝐴_𝑘} is Wijsman strongly

Cesaro summable to𝐴 if {𝑑(𝑥, 𝐴_𝑘)} is strongly summable to

{𝑑(𝑥, 𝐴)}; that is, for each 𝑥 ∈ 𝑋, lim 𝑛 → ∞ 1 𝑛 𝑛 ∑ 𝑘=1󵄨󵄨󵄨󵄨𝑑(𝑥,𝐴𝑘) − 𝑑 (𝑥, 𝐴)󵄨󵄨󵄨󵄨 = 0, (12) (see [7]).

closed subsets 𝐴_𝑘, 𝐵_𝑘 ⊂ 𝑋 such that 𝑑(𝑥, 𝐴_𝑘) > 0 and

𝑑(𝑥, 𝐵_𝑘) > 0 for each 𝑥 ∈ 𝑋, one says that the sequences

{𝐴_𝑘} and {𝐵_𝑘} are asymptotically equivalent (Wijsman sense)

if for each𝑥 ∈ 𝑋, lim 𝑘 → ∞ 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵𝑘) = 1, (13) (denoted by𝐴_𝑘 ∼ 𝐵_𝑘) (see [11]).

As an example, consider the following sequences of circles

in the(𝑥, 𝑦)-plane. 𝐴_𝑘 = {(𝑥, 𝑦) ∈ R2: 𝑥2+ 𝑦2+ 2𝑘𝑦 = 0} , 𝐵_𝑘= {(𝑥, 𝑦) ∈ R2: 𝑥2+ 𝑦2− 2𝑘𝑦 = 0} . (14) Since lim 𝑘 → ∞ 𝑑 (𝑥, 𝐴𝑘) 𝑑 (𝑥, 𝐵_𝑘) = 1, (15)

the sequences{𝐴_𝑘} and {𝐵_𝑘} are asymptotically equivalent

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Definition 10. Let(𝑋, 𝑑) be a metric space. For non-empty

𝑑(𝑥, 𝐵_𝑘) > 0 for each 𝑥 ∈ 𝑋, one says that the sequences {𝐴_𝑘}

and{𝐵_𝑘} are asymptotically statistically equivalent (Wijsman

sense) of multiple𝐿 provided that for every 𝜀 > 0 and for each

𝑥 ∈ 𝑋, lim 𝑛 → ∞ 1 𝑛󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨{𝑘 ≤ 𝑛 :󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨= 0, (16)

(denoted by𝐴_𝑘 𝑊𝑆𝐿∼ 𝐵_𝑘) and simply asymptotically

statisti-cally equivalent (Wijsman sense) if𝐿 = 1 (see [11]).

2. Main Results

Definition 11 (see [12]). Let(𝑋, 𝑑) be a metric space and let

𝐼 ⊆ 2N _{be a proper ideal in} _{N. For any non-empty closed}

subsets𝐴, 𝐴_𝑘⊂ 𝑋, we say that the sequence {𝐴_𝑘} is Wijsman

𝐼-convergent to 𝐴, if for each 𝜀 > 0 and for each 𝑥 ∈ 𝑋, the set

𝐴 (𝑥, 𝜀) = {𝑘 ∈ N : 󵄨󵄨󵄨󵄨𝑑 (𝑥, 𝐴𝑘) − 𝑑 (𝑥, 𝐴)󵄨󵄨󵄨󵄨 ≥𝜀} , (17)

belongs to𝐼. In this case one writes 𝐼_𝑊−lim 𝐴_𝑘= 𝐴 or 𝐴_𝑘 →

𝐴(𝐼_𝑊), and the set of Wijsman 𝐼-convergent sequences of sets

will be denoted by

𝐼_𝑊= {{𝐴_𝑘_{} : {𝑘 ∈ N : 󵄨󵄨󵄨󵄨𝑑 (𝑥, 𝐴}_𝑘_{) − 𝑑 (𝑥, 𝐴)󵄨󵄨󵄨󵄨 ≥𝜀} ∈ 𝐼} . (18)}

As an example, consider the following sequence. Let𝑋 = R2,

and let{𝐴_𝑘} be the following sequence:

𝐴_𝑘 = {{(𝑥, 𝑦) ∈ R2: 𝑥2+ 𝑦2− 2𝑘𝑦 = 0} if 𝑘 ̸= 𝑛2

{(𝑥, 𝑦) ∈ R2_{: 𝑦 = −1}} _if_{𝑘 = 𝑛}2_, (19)

and𝐴 = {(𝑥, 𝑦) ∈ R2 : 𝑦 = 0}. The sequence {𝐴_𝑘} is not

Wijsman convergent to the set𝐴. But, if we take 𝐼 = 𝐼_𝑑, then

{𝐴_𝑘} is Wijsman 𝐼-convergent to set 𝐴, where 𝐼_𝑑is the ideal

of sets that have zero density.

Definition 12. Let (𝑋, 𝑑) be a metric space. For any

non-empty closed subsets𝐴, 𝐴_𝑘 ⊆ 𝑋, we say that the sequence

{𝐴_𝑘} is said to be Wijsman 𝜆-statistically convergent or 𝑊𝑆_𝜆

-convergent to𝐴 if for every 𝜀 > 0 and for each 𝑥 ∈ 𝑋,

lim 𝑛 → ∞

1

𝜆_𝑛󵄨󵄨󵄨󵄨{𝑘 ∈ 𝐼𝑛: 󵄨󵄨󵄨󵄨𝑑 (𝑥, 𝐴𝑘) − 𝑑 (𝑥, 𝐴)󵄨󵄨󵄨󵄨 ≥𝜀}󵄨󵄨󵄨󵄨 = 0. (20)

In this case one writes𝑆_𝜆−lim_𝑊𝐴_𝑘 = 𝐴, or {𝐴_𝑘} → 𝐴(𝑊𝑆_𝜆)

and,

𝑊𝑆_𝜆:= {{𝐴_𝑘} : 𝐴 ⊆ 𝑋, 𝑊𝑆_𝜆− lim 𝐴_𝑘= 𝐴} . (21)

If𝜆_𝑛= 𝑛, then Wijsman 𝜆-statistical convergence is the same

as Wijsman statistical convergence for the sequences of sets.

closed subsets𝐴, 𝐴_𝑘 ⊆ 𝑋, we say that the sequence {𝐴_𝑘} is

said to be Wijsman strongly(𝑉, 𝜆) summable to 𝐴 if for every

𝜀 > 0 and for each 𝑥 ∈ 𝑋, lim

𝑛 → ∞ 1

𝜆_𝑛_{𝑘∈𝐼𝑛}∑󵄨󵄨󵄨󵄨𝑑(𝑥,𝐴𝑘) − 𝑑 (𝑥, 𝐴)󵄨󵄨󵄨󵄨 = 0. (22)

In this case one writes{𝐴_𝑘} → 𝐴[𝑉, 𝜆].

If 𝜆_𝑛 = 𝑛, then [𝑉, 𝜆]-summability reduces to [𝐶,

1]-summability for sequences of sets.

Theorem 14. Let 𝜆 ∈ Λ, (𝑋, 𝑑) be a metric space. For any

non-empty closed subsets𝐴, 𝐴_𝑘⊆ 𝑋, then

(i){𝐴_𝑘} → 𝐴[𝑉, 𝜆] ⇒ {𝐴_𝑘} → 𝐴(𝑊𝑆_𝜆) and the

inclusion[𝑉, 𝜆] ⫅ (𝑊𝑆_𝜆) is proper for sequences of sets,

(ii) if{𝐴_𝑘} is bounded (i.e., {𝐴_𝑘} ∈ 𝐿_∞) and {𝐴_𝑘} →

𝐴(𝑊𝑆_𝜆), then {𝐴_𝑘} → 𝐴[𝑉, 𝜆],

(iii)𝑊𝑆_𝜆∩ 𝐿_∞= [𝑉, 𝜆] ∩ 𝐿_∞,

where𝐿_∞denotes the set of bounded sequences of sets.

Proof. (i) Let𝜀 > 0 and {𝐴_𝑘} → 𝐴[𝑉, 𝜆]. One has

∑ 𝑘∈𝐼𝑛 󵄨󵄨󵄨󵄨𝑑(𝑥,𝐴𝑘) − 𝑑 (𝑥, 𝐴)󵄨󵄨󵄨󵄨 ≥ ∑ 𝑘∈𝐼𝑛 |𝑑(𝑥,𝐴𝑘)−𝑑(𝑥,𝐴)|≥𝜀 󵄨󵄨󵄨󵄨𝑑(𝑥,𝐴𝑘) − 𝑑 (𝑥, 𝐴)󵄨󵄨󵄨󵄨 ≥ 𝜀 ⋅ 󵄨󵄨󵄨󵄨{𝑘 ∈ 𝐼𝑛: 󵄨󵄨󵄨󵄨𝑑 (𝑥, 𝐴𝑘) − 𝑑 (𝑥, 𝐴)󵄨󵄨󵄨󵄨 ≥𝜀}󵄨󵄨󵄨󵄨. (23) Therefore{𝐴_𝑘} → 𝐴[𝑉, 𝜆] ⇒ {𝐴_𝑘} → 𝐴(𝑊𝑆_𝜆).

The following example shows that (𝑊𝑆_𝜆) ⫋ [𝑉, 𝜆] for

sequences of sets:

𝐴_𝑘= {{𝑘} , for 𝑛 − [√𝜆𝑛] + 1 ≤ 𝑘 ≤ 𝑛,

{0} , otherwise. (24)

Then{𝐴_𝑘} ∉ 𝐿_∞and for every𝜀 (0 < 𝜀 ≤ 1),

1

𝜆_𝑛󵄨󵄨󵄨󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨󵄨𝑑 (𝑥, 𝐴𝑘) − 𝑑 (𝑥, {0})󵄨󵄨󵄨󵄨 ≥𝜀}󵄨󵄨󵄨󵄨

= [√𝜆𝑛]

𝜆_𝑛 󳨀→ 0, as 𝑛 󳨀→ ∞,

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that is,{𝐴_𝑘} → {0} (𝑊𝑆_𝜆). On the other hand,

1

𝜆_𝑛_{𝑘∈𝐼𝑛}∑󵄨󵄨󵄨󵄨𝑑(𝑥,𝐴𝑘) − 𝑑 (𝑥, {0})󵄨󵄨󵄨󵄨 󴀀󴀂󴀠 0, as 𝑛 󳨀→ ∞, (26)

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(ii) Suppose that{𝐴_𝑘} is bounded and {𝐴_𝑘} → 𝐴(𝑊𝑆_𝜆).

Then there is a𝑀 such that |𝑑(𝑥, 𝐴_𝑘) − 𝑑(𝑥, 𝐴)| ≤ 𝑀 for all

𝑘. Given 𝜀 > 0, one has 1 𝜆_𝑛_{𝑘∈𝐼𝑛}∑󵄨󵄨󵄨󵄨𝑑(𝑥,𝐴𝑘) − 𝑑 (𝑥, 𝐴)󵄨󵄨󵄨󵄨 =_𝜆1 𝑛 𝑘∈𝐼𝑛∑ |𝑑(𝑥,𝐴𝑘)−𝑑(𝑥,𝐴)|≥𝜀 󵄨󵄨󵄨󵄨𝑑(𝑥,𝐴𝑘) − 𝑑 (𝑥, 𝐴)󵄨󵄨󵄨󵄨 +_𝜆1 𝑛 𝑘∈𝐼𝑛∑ |𝑑(𝑥,𝐴𝑘)−𝑑(𝑥,𝐴)|<𝜀 󵄨󵄨󵄨󵄨𝑑(𝑥,𝐴𝑘) − 𝑑 (𝑥, 𝐴)󵄨󵄨󵄨󵄨 ≤𝑀 𝜆_𝑛 󵄨󵄨󵄨󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨󵄨𝑑 (𝑥, 𝐴𝑘) − 𝑑 (𝑥, 𝐴)󵄨󵄨󵄨󵄨 ≥𝜀}󵄨󵄨󵄨󵄨 + 𝜀, (27)

which implies that{𝐴_𝑘} → 𝐴[𝑉, 𝜆].

(iii) This immediately follows from (i) and (ii).

Definition 15. Let(𝑋, 𝑑) be a metric space, and let 𝐼 be an

admissible ideal. For non-empty closed subsets𝐴_𝑘, 𝐵_𝑘 ⊂ 𝑋

such that 𝑑(𝑥, 𝐴_𝑘) > 0 and 𝑑(𝑥, 𝐵_𝑘) > 0 for each 𝑥 ∈

𝑋, one says that the sequences {𝐴_𝑘} and {𝐵_𝑘} are said to

be asymptotically Wijsman𝐼-equivalent of multiple 𝐿 if for

every𝜀 > 0 and for each 𝑥 ∈ 𝑋,

{𝑘 ∈ N :󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨}

󵄨

𝑑 (𝑥, 𝐴𝑘)

𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀} ∈ 𝐼. (28)

This will be denoted by𝐴_𝑘𝐼𝑊∼ 𝐵_𝑘.

Definition 16. Let(𝑋, 𝑑) be a metric space, and let 𝐼 be an

admissible ideal. For non-empty closed subsets𝐴_𝑘, 𝐵_𝑘 ⊂ 𝑋

such that𝑑(𝑥, 𝐴_𝑘) > 0 and 𝑑(𝑥, 𝐵_𝑘) > 0 for each 𝑥 ∈ 𝑋,

one says that the sequences {𝐴_𝑘} and {𝐵_𝑘} are said to be

strong Cesaro𝐼-asymptotically equivalent (Wijsman sense)

of multiple𝐿 if every 𝜀 > 0 and for each 𝑥 ∈ 𝑋,

{𝑛 ∈ N : 1 𝑛 𝑛 ∑ 𝑘=1 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥ 𝜀} ∈ 𝐼. (29)

This will be denoted by𝐴_𝑘𝐶

𝐿

1(𝐼𝑊)_{∼ 𝐵}

𝑘.

Definition 17. Let (𝑋, 𝑑) be a metric space. For non-empty

𝑑(𝑥, 𝐵𝑘) > 0 for each 𝑥 ∈ 𝑋, one says that the sequences

{𝐴𝑘} and {𝐵𝑘} are Wijsman 𝐼-asymptotically statistically

equivalent of multiple𝐿 if for every 𝜀, 𝛿 > 0 and for each

𝑥 ∈ 𝑋, {𝑛 ∈ N :_𝑛1󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨} 󵄨{𝑘 ≤ 𝑛 : 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝛿} ∈ 𝐼. (30)

This will be denoted by𝐴_𝑘𝑆

𝐿_(𝐼𝑊)

∼ 𝐵_𝑘.

Example 18. Let𝐼 ⊆ 2Nbe a proper ideal inN, and let (𝑋, 𝑑)

be a metric space, then𝐴, 𝐴_𝑘 ⊂ 𝑋 are non-empty closed

subsets. Let𝑋 = R2,{𝐴_𝑘}, {𝐵_𝑘} be the following sequences:

𝐴_𝑘 ={_{ { {(𝑥, 𝑦) ∈ R2_{: 0≤ 𝑥≤𝑛, 0≤𝑦≤}1 𝑛⋅ 𝑥} , if, 𝑘 ̸= 𝑛2, {0, 0} , otherwise, 𝐵_𝑘 = {{(𝑥, 𝑦) ∈ R2: 0 ≤ 𝑥 ≤ 𝑛, 0 ≤ 𝑦 ≤ −1𝑛⋅ 𝑥} , if, 𝑘 ̸= 𝑛2, {0, 0} , otherwise. (31) If we take𝐼 = 𝐼_𝑑we have {𝑘 ∈ N :󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨} 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 1󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀} ∈ 𝐼. (32)

Thus, the sequences {𝐴_𝑘} and {𝐵_𝑘} are asymptotically

𝐼-equivalent (Wijsman sense); that is,𝐴_𝑘𝐼𝑊1

∼ 𝐵𝑘, where𝐼𝑑is the

ideal of sets that have zero density.

Example 19. Let𝐼 ⊆ 2N_{be a proper ideal in}_{N and let, (𝑋, 𝑑) be}

a metric space, then𝐴, 𝐴_𝑘⊂ 𝑋 are non-empty closed subsets.

Let𝑋 = R2,{𝐴_𝑘}, {𝐵_𝑘} be the following sequences:

𝐴_𝑘={_{ { {(𝑥, 𝑦) ∈ R2_{: 𝑥}2_{+ (𝑦 − 1)}2₌ 1 𝑘} , if 𝑘 ̸= 𝑛2 {0, 0} , otherwise, 𝐵𝑘= { { { {(𝑥, 𝑦) ∈ R2: 𝑥2+ (𝑦 + 1)2= 1_𝑘} , if 𝑘 ̸= 𝑛2 {0, 0} , otherwise. (33) If we take𝐼 = 𝐼_𝑑we have {𝑘 ∈ N :󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨} 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵𝑘) − 1 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨≥𝜀} ∈ 𝐼. (34)

Thus, the sequences {𝐴_𝑘} and {𝐵_𝑘} are asymptotically

𝐼-equivalent (Wijsman sense); that is,𝐴_𝑘 𝐼

1 𝑊

∼ 𝐵_𝑘, where𝐼_𝑑 is

the ideal of sets which have zero density.

{𝐴_𝑘} and {𝐵_𝑘} are strongly 𝜆_𝐼-asymptotically equivalent

(Wijsman sense) of multiple𝐿 if for every 𝜀 > 0 and for each

𝑥 ∈ 𝑋, { { { 𝑛 ∈ N : 1 𝜆𝑛𝑘∈𝐼∑𝑛 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥ 𝜀 } } } ∈ 𝐼. (35)

This will be denoted by𝐴_𝑘𝑉

𝐿

𝜆_{∼ 𝐵}(𝐼𝑊)

(5)

{𝐴_𝑘} and {𝐵_𝑘} are 𝐼-asymptotically 𝜆-statistically equivalent

(Wijsman sense) of multiple𝐿, provided that for every 𝜀, 𝛿 >

0 and for each 𝑥 ∈ 𝑋,

{𝑛 ∈ N :_𝜆1 𝑛 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨{𝑘 ∈ 𝐼𝑛: 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝛿} ∈ 𝐼. (36)

This will be denoted by𝐴_𝑘𝑆

𝐿

𝜆(𝐼𝑊)_{∼ 𝐵}

𝑘.

Theorem 22. Let 𝜆 ∈ Λ and let 𝐼 be an admissible ideal in N.

If𝐴_𝑘𝑉 𝐿 𝜆_{∼ 𝐵}(𝐼𝑊) 𝑘, then𝐴𝑘 𝑆𝐿 𝜆(𝐼𝑊)_{∼ 𝐵} 𝑘.

Proof. Assume that𝐴_𝑘𝑉

𝐿 𝜆_{∼ 𝐵}(𝐼𝑊) 𝑘and𝜀 > 0. Then, ∑ 𝑘∈𝐼𝑛 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥ _𝑘∈𝐼∑ 𝑛 |𝑑(𝑥,𝐴𝑘)−𝑑(𝑥,𝐴)|≥𝜀 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 ≥ 𝜀󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨} 󵄨{𝑘 ∈ 𝐼𝑛: 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨, (37) and so, 1 𝜀𝜆_𝑛_{𝑘∈𝐼𝑛}∑󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 ≥ _𝜆1 𝑛 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨{𝑘 ∈ 𝐼𝑛: 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨. (38)

Then for any𝛿 > 0,

{𝑛 ∈ N :_𝜆1 𝑛 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝛿} ⊆{_{ { 𝑛 ∈ N : 1 𝜆𝑛𝑘∈𝐼∑𝑛 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀𝛿 } } } . (39)

Since right hand belongs to𝐼, then left hand also belongs to

𝐼, and this completes the proof.

If{𝐴_𝑘} and {𝐵_𝑘} are bounded and 𝐴_𝑘𝑆

𝐿 𝜆(𝐼𝑊) ∼ 𝐵_𝑘, then𝐴_𝑘𝑉 𝐿 𝜆_∼(𝐼𝑊) 𝐵𝑘.

Proof. Let{𝐴_𝑘}, {𝐵_𝑘} be bounded sequences and let 𝐴_𝑘𝑆

𝐿

𝜆(𝐼𝑊)_∼

𝐵_𝑘. Then there is an𝑀 such that

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨

𝑑 (𝑥, 𝐴_𝑘)

𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≤ 𝑀, (40)

for all𝑘. For each 𝜀 > 0,

1 𝜆_𝑛_{𝑘∈𝐼𝑛}∑󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 = 1 𝜆_𝑛 _{𝑘∈𝐼𝑛}∑ |𝑑(𝑥,𝐴𝑘)−𝑑(𝑥,𝐴)|≥𝜀 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 + 1 𝜆_𝑛 _{𝑘∈𝐼𝑛}∑ |𝑑(𝑥,𝐴𝑘)−𝑑(𝑥,𝐴)|<𝜀 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 ≤ 𝑀 1 𝜆_𝑛󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨{𝑘 ∈ 𝐼𝑛:󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥ 𝜀 2}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨+ 𝜀 2. (41) Then, { { { 𝑛 ∈ N : 1 𝜆𝑛𝑘∈𝐼∑𝑛 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥ 𝜀 } } } ⊆ {𝑛 ∈ N : 1 𝜆_𝑛 ×󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨} 󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥ 𝜀 2}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥ 𝜀 2𝑀} ∈ 𝐼. (42) Therefore,𝐴_𝑘𝑉 𝐿 𝜆_{∼ 𝐵}(𝐼𝑊) 𝑘.

The following example shows that if{𝐴_𝑘} and {𝐵_𝑘} are not

bounded, thenTheorem 23cannot be true.

Example 24. Take𝐿 = 1 and define {𝐴_𝑘} to be,

𝐴_𝑘= {{𝑘} , 𝑘 = 𝑘𝑟−1+ 1, 𝑘𝑟−1+ 2, . . . , 𝑘𝑟−1+ [√𝜆𝑛]

{1} , otherwise,

(43)

where⌊⋅⌋ denotes the greatest integer function and 𝐵_𝑘 = {1}

for all𝑘. Note that {𝐴_𝑘} is not bounded. Then 𝐴_𝑘 𝑆

𝐿

𝜆_{∼ 𝐵}(𝐼)

𝑘, but

𝐴_𝑘𝑉𝜆𝐿_{∼ 𝐵}(𝐼)

𝑘is not true.

If𝐴_𝑘𝑉 𝐿 𝜆_{∼ 𝐵}(𝐼𝑊) 𝑘, then𝐴𝑘 𝐶𝐿 1_{∼ 𝐵}(𝐼𝑊) 𝑘.

(6)

Proof. Assume that𝐴_𝑘𝑉 𝐿 𝜆_{∼ 𝐵}(𝐼) 𝑘and𝜀 > 0. Then, 1 𝑛 𝑛 ∑ 𝑘=1 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵𝑘) − 𝐿 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 = 1 𝑛 𝑛−𝜆𝑛 ∑ 𝑘=1 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨+ 1 𝑛_{𝑘∈𝐼𝑛}∑󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 ≤ 1 𝜆𝑛 𝑛−𝜆𝑛 ∑ 𝑘=1 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨+ 1 𝜆𝑛_{𝑘∈𝐼𝑛}∑ 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 ≤ _𝜆2 𝑛𝑘∈𝐼𝑛∑ 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨, (44) and so, {𝑛 ∈ N : 1_𝑛∑𝑛 𝑘=1 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀} ⊆ {𝑛 ∈ N : _𝜆1 𝑛 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥ 𝜀 2} ∈ 𝐼. (45) Hence𝐴_𝑘𝐶 𝐿 1(𝐼𝑊) ∼ 𝐵_𝑘.

Theorem 26. If lim inf 𝜆_𝑛/𝑛 > 0, then 𝐴_𝑘 𝑆𝐿(𝐼𝑊)∼ 𝐵_𝑘implies

𝐴_𝑘𝑆𝐿𝜆(𝐼𝑊)_{∼ 𝐵}

𝑘.

Proof. Assume that lim inf(𝜆_𝑛/𝑛) > 0 and there exists a 𝛿 > 0

such that𝜆_𝑛/𝑛 ≥𝛿 for sufficiently large 𝑛. For given 𝜀 > 0 one

has, 1 𝑛{𝑘 ≤ 𝑛 :󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵𝑘) − 𝐿 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨≥𝜀} ⊇ 1 𝑛{𝑘 ∈ 𝐼𝑛 :󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀} . (46) Therefore, 1 𝑛󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨{𝑘 ≤ 𝑛 :󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 ≥ 1_𝑛󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨} 󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵𝑘) − 𝐿 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨≥𝜀} 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 ≥ 𝜆𝑛 𝑛 1 𝜆_𝑛󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨{𝑘 ∈ 𝐼𝑛 :󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 𝑑 (𝑥, 𝐴𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 ≥ 𝛿1 𝜆_𝑛 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨{𝑘 ∈ 𝐼𝑛:󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨; (47)

then for any𝜂 > 0 we get

{𝑛 ∈ N : _𝜆1 𝑛 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨{𝑘 ∈ 𝐼𝑛: 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜂} ⊆ {𝑛 ∈ N : 1 𝑛󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨{𝑘 ≤ 𝑛 :󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 𝑑 (𝑥, 𝐴_𝑘) 𝑑 (𝑥, 𝐵_𝑘) − 𝐿󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜀}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≥𝜂𝛿} ∈ 𝐼, (48) and this completes the proof.

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