On Lacunary Ideal Convergence in Random 𝑛-Normed Space

Abstract

We introduce the notions of lacunary ℐ-convergence and lacunary ℐ-Cauchy in the topology induced by random 𝑛-normed spaces and prove some important results.

1. Introduction

Menger [1] generalized the metric axioms by associating a distribution function with each pair of points of a set. This system, called a probabilistic metric space, originally a statistical metric space, has been developed extensively by Schweizer and Sklar [2, 3]. The idea of Menger was to use distribution function instead of nonnegative real numbers as values of the metric, which was further developed by several other authors. In this theory, the notion of distance has a probabilistic nature. Namely, the distance between two points 𝑥 and 𝑦 is represented by a distribution function 𝐹_𝑥𝑦, and for 𝑡>0, the value 𝐹_𝑥𝑦(𝑡) is interpreted as the probability that the distance from 𝑥 to 𝑦 is less than 𝑡. Using this concept, Šerstnev [4] introduced the concept of probabilistic normed spaces. It provides an important area into which many deterministic results of linear normed spaces can be generalized. The studies of continuity properties, linear operators, statistical convergence, and ideal convergence in probabilistic normed spaces have gained many attractions, and such studies have diverse applications into various fields [5–13].

In [14], Gähler introduced an attractive theory of 2-normed and 𝑛-normed spaces in the 1960s. Since then, many researchers have studied these subjects and obtained various results [15–19].

Since the introduction of the notion of statistical convergence for sequences of real numbers by Steinhaus [20] and Fast [21] independently, the theory has been investigated and developed by several authors. There has been an effort to introduce several generalizations and variants of statistical convergence in different spaces. One very important generalization of this notion was introduced by Kostyrko et al. [22] by using an ideal ℐ of subsets of the set of natural numbers, which they called ℐ-convergence.

Another important variant of statistical convergence is the notion of lacunary statistical convergence introduced by Fridy and Orhan [23]. Recently, Mohiuddine and Aiyub [24] studied lacunary statistical convergence by introducing the concept 𝜃-statistical convergence in random 2-normed space. Their work can be considered as a particular generalization of the statistical convergence. In [25], Mursaleen and Mohiuddine extended the idea of lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, and Debnath [26] investigated lacunary ideal convergence in intuitionistic fuzzy normed linear spaces. Also, lacunary statistically convergent double sequences in probabilistic normed space were studied by Mohiuddine and Savaş in [27].

The notion of lacunary ideal convergence has not been studied previously in the setting of 𝑛-normed linear spaces. Motivated by this fact, in this paper, as a variant of ℐ-convergence, the notion of lacunary ideal convergence is introduced in a random 𝑛-normed space, and some important results are established. Finally, the notions of lacunary ℐ^𝐹-Cauchy and lacunary ℐ^𝐹∗-Cauchy sequences are introduced and studied.

Throughout the paper, ℕ will denote the set of all natural numbers. First, we recall some of the basic concepts which will be used in this paper.

Definition 1 (see [15]). Let 𝑛∈ℕ, and let 𝑋 be a real vector space of dimension 𝑑≥𝑛, where 𝑛≤𝑑<∞. A real-valued function ‖⋅,…,⋅‖ on 𝑋×𝑋×⋯×𝑋=𝑋^𝑛, satisfying the following properties:(i)‖𝑥₁,𝑥₂,…,𝑥_𝑛‖=0 if and only if 𝑥₁,𝑥₂,…,𝑥_𝑛 are linearly dependent,(ii)‖𝑥₁,𝑥₂,…,𝑥_𝑛‖ is invariant under any permutation of 𝑥₁,𝑥₂,…,𝑥_𝑛,(iii)‖𝑥₁,𝑥₂,…,𝛼𝑥_𝑛‖=|𝛼|‖𝑥₁,𝑥₂,…,𝑥_𝑛‖, where 𝛼∈ℝ (set of real numbers),(iv)‖𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑦+𝑧‖≤‖𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑦‖+‖𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑧‖, is called an 𝑛-norm on 𝑋, and the pair (𝑋,‖⋅,…,⋅‖) is called an 𝑛-normed space.

Given an 𝑛-normed space (𝑋,‖⋅,…,⋅‖), one can derive a topology for it via the following definition of the limit of a sequence: a sequence 𝑥(𝑘) in 𝑋 is said to be convergent to 𝑥 in 𝑋 if lim_𝑘→∞‖𝑥₁,…,𝑥_𝑛−1,𝑥(𝑘)−𝑥‖=0 for every 𝑥₁,…,𝑥_𝑛−1∈𝑋.

All the concepts listed later are studied in depth in the fundamental book by Schweizer and Sklar [3].

Definition 2. Let ℝ denote the set of real numbers, ℝ₊={𝑥∈ℝ∶𝑥≥0}, and 𝑆=[0,1] the closed unit interval. A mapping 𝑓∶ℝ→𝑆 is called a distribution function if it is nondecreasing and left continuous with inf_𝑡∈ℝ𝑓(𝑡)=0 and sup_𝑡∈ℝ𝑓(𝑡)=1.

We denote the set of all distribution functions by 𝐷⁺ such that 𝑓(0)=0. If 𝑎∈ℝ₊, then 𝐻_𝑎∈𝐷⁺, where 𝐻_𝑎(𝑡)=1if0𝑡>𝑎,if𝑡≤𝑎. (1) It is obvious that 𝐻₀≥𝑓 for all 𝑓∈𝐷⁺.

Definition 3. A triangular norm (𝑡-norm) is a continuous mapping ∗∶𝑆×𝑆→𝑆 such that (𝑆,∗) is an abelian monoid with unit one and 𝑐∗𝑑≤𝑎∗𝑏 if 𝑐≤𝑎 and 𝑑≤𝑏 for all 𝑎,𝑏,𝑐,𝑑∈𝑆. A triangle function 𝜏 is a binary operation on 𝐷⁺ which is commutative and associative, and 𝜏(𝑓,𝐻₀)=𝑓 for every 𝑓∈𝐷⁺.

Definition 4 (see [28]). Let 𝑋 be a linear space of dimension greater than one, 𝜏 a triangle function, and 𝐹 a mapping from 𝑋×𝑋×⋯×𝑋_𝑛 into 𝐷⁺. Then, 𝐹 is called a probabilistic 𝑛-norm and (𝑋,𝐹,𝜏) a probabilistic 𝑛-normed space if the following conditions are satisfied:(i)𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛;𝑡)=𝐻₀(𝑡) if and only if 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 are linearly dependent, where 𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛;𝑡) denotes the value of 𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛) at 𝑡∈ℝ;(ii)𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛;𝑡)≠𝐻₀(𝑡) if and only if 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 are linearly independent;(iii)𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛;𝑡) is invariant under any permutation 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋;(iv)𝐹(𝑥₁,𝑥₂,…,𝛼𝑥_𝑛;𝑡)=𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛;𝑡/|𝛼|) for every 𝑡>0, 𝛼≠0 and 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋;(v)𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛+𝑥^′_𝑛;𝑡)≥𝜏(𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛;𝑡),𝐹(𝑥₁,𝑥₂,…,𝑥^′_𝑛;𝑡)) whenever 𝑥₁,𝑥₂,…,𝑥_𝑛,𝑥^′_𝑛∈𝑋, and 𝑡>0.If (v) is replaced by (vi)𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛+𝑥^′_𝑛;𝑡₁+𝑡₂)≥𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛;𝑡₁)∗𝐹(𝑥₁,𝑥₂,…,𝑥^′_𝑛;𝑡₂) for all 𝑥₁,𝑥₂,…,𝑥_𝑛,𝑥^′_𝑛∈𝑋 and 𝑡₁,𝑡₂∈ℝ₊,then (𝑋,𝐹,∗) is called a random 𝑛-normed space (for short, RnN space).

Remark 5. Note that every 𝑛-norm space (𝑋,‖⋅,…,⋅‖) can be made a random 𝑛-normed space in a natural way, by setting(i)𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛;𝑡)=𝐻₀(𝑡−‖𝑥₁,𝑥₂,…,𝑥_𝑛‖), for every 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋, 𝑡>0 and 𝑎∗𝑏=min{𝑎,𝑏}, 𝑎,𝑏∈𝑆;(ii)𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛;𝑡)=𝑡/(𝑡+‖𝑥₁,𝑥₂,…,𝑥_𝑛‖) for every 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋, 𝑡>0 and 𝑎∗𝑏=𝑎𝑏 for 𝑎,𝑏∈𝑆.

Let (𝑋,𝐹,∗) be a RnN space. Since ∗ is a continuous 𝑡-norm, the system of (𝜀,𝜆)-neighborhoods of 𝜙 (the null vector in 𝑋) 𝒩_𝜙(𝜀,𝜆)∶𝜀>0,𝜆∈(0,1), (2) where 𝒩_𝜙(𝜀,𝜆)=𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋∶𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1}(𝜀)>1−𝜆, (3) determines a first countable Hausdorff topology on 𝑋, called the 𝐹-topology. Thus, the 𝐹-topology can be completely specified by means of 𝐹-convergence of sequences. It is clear that 𝑥₁,𝑥₂,…,𝑥_𝑛−1−𝑦∈𝒩_𝜙 means that 𝑦∈𝒩_{𝑥1,𝑥2,…,𝑥𝑛−1} and vice versa.

A sequence 𝑥=(𝑥_𝑘) in 𝑋 is said to have 𝐹-convergence to 𝐿∈𝑋 if for every 𝜀>0, 𝜆∈(0,1) and for each nonzero 𝑥₁,…,𝑥_𝑛−1∈𝑋 there exists a positive integer 𝑁 such that 𝑥₁,…,𝑥_𝑛−1,𝑥_𝑘−𝐿∈𝒩_𝜙(𝜀,𝜆)foreach𝑛≥𝑁, (4) or equivalently, 𝑥₁,…,𝑥_𝑛−1,𝑥_𝑘∈𝒩_𝐿(𝜀,𝜆)foreach𝑛≥𝑁. (5) In this case, we write 𝐹-lim𝑥₁,…,𝑥_𝑛−1,𝑥_𝑘=𝐿.

Lemma 6. Let (𝑋,‖⋅,…,⋅‖) be a real 𝑛-normed space, and let (𝑋,𝐹,∗) be RnN space induced by the random norm 𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛;𝑡)=𝑡/(𝑡+‖𝑥₁,𝑥₂,…,𝑥_𝑛‖), where 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑡>0. Then, for every sequence 𝑥=(𝑥_𝑘) and nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1 in 𝑋, lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥−𝐿=0⇒𝐹−lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,(𝑥−𝐿)=0. (6)

Proof. Suppose that lim‖𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥−𝐿‖=0. Then, for every 𝑡>0 and for every 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋, there exists a positive integer 𝑁=𝑁(𝑡) such that 𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘−𝐿<𝑡foreach𝑘>𝑁. (7) We observe that for any given 𝜀>0, 𝜀+𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘−𝐿𝜀<𝜀+𝑡𝜀 (8) which is equivalent to 𝜀𝜀+𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘>𝜀−𝐿𝑡𝜀+𝑡=1−.𝜀+𝑡 (9) Therefore, by letting 𝜆=𝑡/(𝜀+𝑡)∈(0,1), we have 𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1,𝑥𝑘−𝐿}(𝜀)>1−𝜆foreach𝑘>𝑁. (10) This implies that 𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∈𝒩_𝐿(𝜀,𝜆) for each 𝑘>𝑁 as desired.

Definition 7 (see [21, 29]). A subset 𝐸 of ℕ is said to have density 𝛿(𝐸) if 𝛿(𝐸)=lim_𝑛→∞𝑛⁻¹∑^𝑛_𝑘=1𝜒_𝐸(𝑘) exists. A number sequence (𝑥_𝑛)_𝑛∈ℕ is said to be statistically convergent to 𝐿 if for every 𝜀>0,𝛿({𝑛∈ℕ∶|𝑥_𝑛−𝐿|≥𝜀})=0. If (𝑥_𝑛)_𝑛∈ℕ is statistically convergent to 𝐿, we write st-lim𝑥_𝑛=𝐿, which is necessarily unique.

Definition 8 (see [22]). A family ℐ⊂2^𝑌 of subsets of a nonempty set 𝑌 is said to be an ideal in 𝑌 if (i) ∅∈ℐ; (ii) 𝐴,𝐵∈ℐ imply 𝐴∪𝐵∈ℐ; (iii) 𝐴∈ℐ,𝐵⊂𝐴 imply 𝐵∈ℐ. A nontrivial ideal ℐ in 𝑌 is called an admissible ideal if it is different from 𝑃(ℕ) and it contains all singletons, that is, {𝑥}∈ℐ for each 𝑥∈𝑌.
Let 𝐼⊂𝑃(𝑌) be a nontrivial ideal. A class ℱ(ℐ)={𝑀⊂𝑌∶∃𝐴∈ℐ∶𝑀=𝑌⧵𝐴}, called the filter associated with the ideal ℐ, is a filter on 𝑌.

Definition 9 (see [22]). Let ℐ⊂2^ℕ be a nontrivial ideal in ℕ. Then, a sequence (𝑥_𝑛)_𝑛∈ℕ in 𝑋 is said to be ℐ-convergent to 𝐿∈𝑋, if for each 𝜀>0 the set 𝐴(𝜀)={𝑛∈ℕ∶‖𝑥_𝑛−𝐿‖≥𝜀} belongs to ℐ.

Definition 10. By a lacunary sequence we mean an increasing integer sequence 𝜃=(𝑘_𝑟) such that 𝑘₀=0 and ℎ_𝑟∶=𝑘_𝑟−𝑘_𝑟−1→∞ as 𝑟→∞. Throughout this paper, the intervals determined by 𝜃 will be denoted by 𝐼_𝑟∶=(𝑘_𝑟−1,𝑘_𝑟]. Let 𝐾⊆ℕ. The number 𝛿_𝜃(𝐾)=lim_𝑟1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑘∈𝐾 (11) is said to be the 𝜃-density of 𝐾, provided the limit exists (see [23]).

Definition 11 (see [30]). Let ℐ⊂2^ℕ be a nontrivial ideal in ℕ. A sequence 𝑥=(𝑥_𝑘) of numbers is said to be lacunary ℐ-convergent to a number 𝐿 if, for every 𝜀>0, 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥_𝑘−𝐿≥𝜀∈ℐ. (12)
In this case, we write ℐ_𝜃-lim𝑥=𝐿.

2. Lacunaryℐ^𝐹andℐ^𝐹∗-Convergence for Sequences in RnN Spaces

In this section, we study the concepts lacunary ℐ^𝐹 and ℐ^𝐹∗-convergence of a sequence in (𝑋,𝐹,∗) and prove some important results.

Definition 12. Let (𝑋,𝐹,∗) be RnN space, and let ℐ be a proper ideal in ℕ. A sequence 𝑥=(𝑥_𝑘) in 𝑋 is said to be ℐ^𝐹-convergent to 𝐿∈𝑋 (ℐ^𝐹-convergent to 𝐿∈𝑋 with respect to the random 𝑛-norm 𝐹-topology) if for each 𝜀>0, 𝜆∈(0,1) and each nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋, 𝑘∈ℕ∶𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∉𝒩_𝐿(𝜀,𝜆)∈ℐ. (13)

In this case, the vector 𝐿 is called the ℐ^𝐹-limit of the sequence 𝑥=(𝑥_𝑘), and we write ℐ^𝐹-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥=𝐿.

Definition 13. Let (𝑋,𝐹,∗) be RnN space, and let ℐ be a proper ideal in ℕ. A sequence 𝑥=(𝑥_𝑘) in 𝑋 is said to be lacunary ℐ^𝐹-convergent to 𝐿∈𝑋 with respect to the random 𝑛-norm 𝐹-topology if for each 𝜀>0, 𝜆∈(0,1) and each nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋, 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∉𝒩_𝐿(𝜀,𝜆)∈ℐ. (14)

In this case, the vector 𝐿 is called the ℐ^𝐹_𝜃-limit of the sequence 𝑥=(𝑥_𝑘), and we write ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥=𝐿.

Lemma 14. Let (𝑋,𝐹,∗) be RnN space. If a sequence 𝑥=(𝑥_𝑘) is ℐ^𝐹_𝜃-convergent with respect to the random 𝑛-norm 𝐹, then ℐ^𝐹_𝜃-limit is unique.

Proof. Let us assume that ℐ^𝐹_𝜃−lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥=𝐿₁,ℐ^𝐹_𝜃−lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥=𝐿₂, (15) where 𝐿₁≠𝐿₂. Since 𝐿₁≠𝐿₂, select 𝜀>0, 𝜆∈(0,1) and each nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋 such that 𝒩_𝐿1(𝜀,𝜆) and 𝒩_𝐿2(𝜀,𝜆) are disjoint neighborhoods of 𝐿₁ and 𝐿₂. Since 𝐿₁ and 𝐿₂ both are ℐ^𝐹_𝜃-limit of the sequence (𝑥_𝑘), we have 1𝐴=𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∉𝒩_𝐿1(),1𝜀,𝜆𝐵=𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∉𝒩_𝐿2(𝜀,𝜆), (16) both belonging to ℐ. This implies that the sets 𝐴^𝑐=1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∈𝒩_𝐿1(𝜀,𝜆) (17) and 𝐵^𝑐={𝑟∈ℕ∶(1/ℎ_𝑟)∑_{𝑘∈𝐼𝑟}(𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∈𝒩_𝐿2(𝜀,𝜆))} belong to ℱ(ℐ). In this way, we obtain a contradiction to the fact that the neighborhoods 𝒩_𝐿1(𝜀,𝜆) and 𝒩_𝐿2(𝜀,𝜆) of 𝐿₁ and 𝐿₂ are disjoints. Hence, we have 𝐿₁=𝐿₂. This completes the proof.

Lemma 15. Let (𝑋,𝐹,∗) be RnN space. Then one has(i)𝐹^𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿, then ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿;(ii)if ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿₁ and ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑦_𝑘=𝐿₂, then ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,(𝑥_𝑘+𝑦_𝑘)=𝐿₁+𝐿₂;(iii)if ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿 and 𝛼∈ℝ, then ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝛼𝑥_𝑘=𝛼𝐿;(iv)if ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿₁, and ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿₂, then ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,(𝑥_𝑘−𝑦_𝑘)=𝐿₁−𝐿₂.

Proof. (i) Suppose that 𝐹^𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿. Let 𝜀>0, 𝜆∈(0,1) and nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋. Then, there exists positive integer 𝑁 such that (1/ℎ_𝑟)∑_{𝑘∈𝐼𝑟}(𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∈𝒩_𝐿(𝜀,𝜆)) for each 𝑘>𝑁. Since the set 1𝐴=𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∉𝒩_𝐿1()⊆{}𝜀,𝜆1,2,3,…,𝑁−1 (18) and the ideal ℐ is admissible, we have 𝐴∈ℐ. This shows that ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿.
(ii) Let 𝜀>0, 𝜆∈(0,1) and nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋. Choose 𝜂∈(0,1) such that (1−𝜂)∗(1−𝜂)>(1−𝜆). Since ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿₁ and ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑦_𝑘=𝐿₂, the sets 1𝐴=𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∉𝒩_𝐿1𝜀2,1,𝜆𝐵=𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑦_𝑘∉𝒩_𝐿2𝜀2,𝜆 (19) belong to ℐ. Let 𝐶={𝑟∈ℕ∶(1/ℎ_𝑟)∑_{𝑘∈𝐼𝑟}(𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘+𝑦_𝑘∉𝒩_𝐿1+𝐿2(𝜀,𝜆))}. Since ℐ is an ideal, it is sufficient to show that 𝐶⊂𝐴∪𝐵. This is equivalent to show that 𝐶^𝑐⊃𝐴^𝑐∩𝐵^𝑐 where 𝐴^𝑐 and 𝐵^𝑐 belong to ℱ(ℐ). Let 𝑟∈𝐴^𝑐∩𝐵^𝑐, that is, 𝑟∈𝐴^𝑐 and 𝑟∈𝐵^𝑐, and we have 𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1,(𝑥𝑘+𝑦𝑘)−(𝐿1+𝐿2)}(𝜀)≥𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1,𝑥𝑘−𝐿1}𝜀2∗𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1,𝑦𝑘−𝐿2}𝜀2>∗>().1−𝜂1−𝜂1−𝜆 (20) Since 𝑟∈𝐶^𝑐⊃𝐴^𝑐∩𝐵^𝑐∈ℱ(ℐ), we have 𝐶⊂𝐴∪𝐵∈ℐ.
(iii) It is trivial for 𝛼=0. Now, let 𝛼≠0, 𝜀>0, 𝜆∈(0,1), and nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋. Since ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿, we have 1𝐴=𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∉𝒩_𝐿(𝜀,𝜆)∈ℐ (21) This implies that 𝐴^𝑐=1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∈𝒩_𝐿()(ℐ).𝜀,𝜆∈ℱ (22) Let 𝑟∈𝐴^𝑐. Then, we have 𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1,𝛼𝑥𝑘−𝛼𝐿}(𝜀)=𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1,𝑥𝑘−𝐿}𝜀|𝛼|≥𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1,𝑥𝑘−𝐿}(𝜀)∗𝐹₀𝜀|𝛼|>()=().−𝜀1−𝜆∗11−𝜆 (23) So, 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝛼𝑥_𝑘∉𝒩_𝛼𝐿(𝜀,𝜆)∈ℐ. (24) Hence, ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝛼𝑥_𝑘=𝛼𝐿.
(iv) The result follows from (ii) and (iii).

We introduce the concept of ℐ^𝐹∗_𝜃-convergence closely related to ℐ^𝐹_𝜃-convergence of sequence in random 𝑛-normed space and show that ℐ^𝐹∗_𝜃-convergence implies ℐ^𝐹_𝜃-convergence but not conversely.

Definition 16. Let (𝑋,𝐹,∗) be RnN space, and let ℐ be an admissible ideal. We say that a sequence 𝑥=(𝑥_𝑘) in 𝑋 is said to be lacunary ℐ^𝐹∗-convergent to 𝐿∈𝑋 with respect to the random 𝑛-norm 𝐹 if there is a set 𝐾={𝑚₁<𝑚₂<⋯<𝑚_𝑘<⋯}⊆ℕ such that 𝐾^′={𝑟∈ℕ∶𝑚_𝑘∈𝐼_𝑟}∈ℱ(ℐ) and 𝐹^𝜃-lim_𝑘𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑚𝑘=𝐿 for each nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋.

In this case, we write ℐ_𝜃^𝐹∗-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿, and 𝐿 is called the ℐ_𝜃^𝐹∗-limit of the sequence 𝑥=(𝑥_𝑘).

Theorem 17. Let (𝑋,𝐹,∗) be RnN space, and let ℐ be an admissible ideal. If ℐ_𝜃^𝐹∗-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿, then ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿.

Proof. Suppose that ℐ_𝜃^𝐹∗-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿. Then, by definition, there exists 𝐾=𝑚₁<𝑚₂<⋯<𝑚_𝑘<⋯⊆ℕ (25) such that 𝐾^′∈ℱ(ℐ) and 𝐹^𝜃-lim_𝑘𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑚𝑘=𝐿 for each nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋. Let 𝜀>0, 𝜆∈(0,1) and nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋 be given. Since 𝐹^𝜃-lim_𝑘𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑚𝑘=𝐿, there exists 𝑁∈ℕ such that 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑚𝑘∈𝒩_𝐿(𝜀,𝜆) (26) for every 𝑚≥𝑁. Since 1𝐴=𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑚𝑘∉𝒩_𝐿(𝜀,𝜆) (27) is contained in 𝐵=𝑚₁,𝑚₂,…,𝑚_𝑁−1 (28) and the ideal ℐ is admissible, we have 𝐴∈ℐ. Hence, 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∉𝒩_𝐿(𝜀,𝜆)⊆𝐾∪𝐵∈ℐ (29) for 𝜀>0, 𝜆∈(0,1) and nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋. Therefore, we conclude that ℐ^𝐹_𝜃-lim𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘=𝐿.

The following example shows that the converse of Theorem 17 needs not to be true.

Example 18. Consider 𝑋=ℝ^𝑛 with 𝑥₁,𝑥₂,…,𝑥_𝑛𝑥=𝑎𝑏𝑠¹¹…𝑥_1𝑛⋮𝑥_𝑛1…𝑥_𝑛𝑛, (30) where 𝑥_𝑖=(𝑥_𝑖1,…,𝑥_𝑖𝑛)∈ℝ^𝑛 for each 𝑖=1,2,…,𝑛, and let 𝑎∗𝑏=𝑎𝑏 for all 𝑎,𝑏∈𝑆. For all (𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥)∈ℝ^𝑛 and 𝑡>0, consider 𝐹_{𝑦1,𝑦2,…,𝑦𝑛−1,𝑥}(𝑡)=𝑡𝑡+𝑦₁,𝑦₂,…,𝑦_𝑛−1.,𝑥 (31) Then, (ℝ^𝑛,𝐹,∗) is RnN space. Consider a decomposition of ℕ as ℕ=∪_𝑖Δ_𝑖 such that for any 𝑚∈ℕ, each Δ_𝑖 contains infinitely many (𝑖)’s, where 𝑖≥𝑚 and Δ_𝑖∩Δ_𝑚=∅ for 𝑖≠𝑚. Let  ℐ be the class of all subsets of ℕ which intersect almost a finite number of Δ_𝑖’s. Then, ℐ is an admissible ideal. We define a sequence (𝑥_𝑚) as follows: 𝑥_𝑚=(1/𝑖,0,0,…,0)∈ℝ^𝑛, 𝑖=1,2,…, if 𝑚∈Δ_𝑖. Then, for nonzero 𝑦₁,𝑦₂,…,𝑦_𝑛−1∈𝑋, we have 𝐹_{𝑥𝑚,𝑦1,𝑦2,…,𝑦𝑛−1}(𝑡)=𝑡𝑡+𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥_𝑚,→1 (32) as 𝑚→∞. Hence, ℐ^𝐹_𝜃-lim_𝑚𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥_𝑚=0.
Now, we show that ℐ^𝐹∗_𝜃-lim_𝑚𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥_𝑚≠0. Suppose that ℐ^𝐹∗_𝜃−lim_𝑚𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥_𝑚=0. (33) Then, by definition, there exists a subset 𝐾=𝑚_𝑗∶𝑚₁<𝑚₂<⋯⊂ℕ (34) such that 𝐾^′∈ℱ(ℐ) and 𝐹^𝜃-lim_𝑗𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥_𝑚𝑗=0. Since 𝐾^′∈ℱ(ℐ), there exists 𝐻∈ℐ such that 𝐾=ℕ⧵𝐻. Then, there exist positive integers 𝑝 such that 𝐻⊂^𝑝_𝑚=1Δ_𝑚. (35) Thus, Δ_𝑝+1⊂𝐾, and so 𝑥_𝑚𝑗=1/(𝑝+1)>0 for infinitely many values 𝑚_𝑗’s in 𝐾. This contradicts the assumption that 𝐹^𝜃-lim_𝑗𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥_𝑚𝑗=0. Hence, ℐ^𝐹∗_𝜃-lim_𝑚𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥_𝑚≠0.

Hence, the converse of the theorem needs not to be true.

The following theorem shows that the converse holds if the ideal ℐ satisfies condition (𝐴𝑃).

Definition 19 (see [22]). An admissible ideal ℐ⊂𝑃 is said to satisfy the condition (𝐴𝑃) if for every sequence (𝐴_𝑛)_𝑛∈ℕ of pairwise disjoint sets from ℐ there are sets 𝐵_𝑛⊂ℕ, 𝑛∈ℕ, such that the symmetric difference 𝐴_𝑛Δ𝐵_𝑛 is a finite set for every 𝑛 and ∪_𝑛∈ℕ𝐵_𝑛∈ℐ.

Theorem 20. Let (𝑋,𝐹,∗) be RnN space, and let the ideal ℐ satisfy the condition (𝐴𝑃). If 𝑥=(𝑥_𝑘) is a sequence in 𝑋 such that ℐ^𝐹_𝜃-lim𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥=𝐿, then ℐ^𝐹∗_𝜃-lim𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥=𝐿.

Proof. Since ℐ^𝐹_𝜃-lim𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥=𝐿, so for every 𝜀>0,𝜆∈(0,1) and nonzero 𝑦₁,𝑦₂,…,𝑦_𝑛−1∈𝑋, the set 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥_𝑘∉𝒩_𝐿(𝜀,𝜆)∈ℐ. (36) We define the set 𝐴_𝑝 for 𝑝∈ℕ as 𝐴_𝑝=1𝑘∈ℕ∶1−𝑝≤𝐹_{𝑦1,𝑦2,…,𝑦𝑛−1,𝑥𝑘−𝐿}1<1−.𝑝+1 (37) Then, it is clear that {𝐴₁,𝐴₂,…} is a countable family of mutually disjoint sets belonging to ℐ, and so by the condition (𝐴𝑃), there is a countable family of sets {𝐵₁,𝐵₂,…}∈ℐ such that the symmetric difference 𝐴_𝑖Δ𝐵_𝑖 is a finite set for each 𝑖∈ℕ and 𝐵=∪^∞_𝑖=1𝐵_𝑖∈ℐ. Since 𝐵∈ℐ, there is a set 𝐾∈ℱ(ℐ) such that 𝐾=ℕ⧵𝐵. Now, we prove that the subsequence (𝑥_𝑘)_𝑘∈𝐾 is convergent to 𝐿 with respect to the random 𝑛-norm 𝐹. Let 𝜂∈(0,1), 𝜀>0 and nonzero 𝑦₁,𝑦₂,…,𝑦_𝑛−1∈𝑋. Choose a positive 𝑞 such that 𝑞⁻¹<𝜂. Then, 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥_𝑘∉𝒩_𝐿⊂1𝜀,𝜂𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥_𝑘∉𝒩_𝐿1𝜀,𝑞⊂^𝑞−1_𝑖=1𝐴_𝑖. (38) Since 𝐴_𝑖Δ𝐵_𝑖 is a finite set for each 𝑖=1,2,…,𝑞−1, there exists 𝑗₀∈ℕ such that ^𝑞−1_𝑖=1𝐵_𝑖∩𝑘∈ℕ∶𝑘≥𝑘₀=^𝑞−1_𝑖=1𝐴_𝑖∩𝑘∈ℕ∶𝑘≥𝑘₀. (39) If 𝑘≥𝑘₀ and 𝑘∈𝐾, then 𝑘∉∪^𝑞−1_𝑖=1𝐵_𝑖 and 𝑘∉∪^𝑞−1_𝑖=1𝐴_𝑖. Hence, for every 𝑘≥𝑘₀ and 𝑘∈𝐾, we have 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥_𝑘∉𝒩_𝐿𝜀,𝜂. (40) Since this holds for every 𝜀>0,  𝜂∈(0,1) and nonzero 𝑦₁,𝑦₂,…,𝑦_𝑛−1∈𝑋, so we have ℐ^𝐹∗_𝜃-lim𝑦₁,𝑦₂,…,𝑦_𝑛−1,𝑥=𝐿. This completes the proof of the theorem.

3. ℐ^𝐹_𝜃 and ℐ^𝐹∗_𝜃-Cauchy Sequences in RnN Spaces

In this section, we study the concepts ℐ^𝐹_𝜃-Cauchy and ℐ^𝐹∗_𝜃-Cauchy of a sequence in (𝑋,𝐹,∗). Also, we will study the relations between these concepts.

Definition 21. Let (𝑋,𝐹,∗) be RnN space, and let ℐ be an admissible ideal of ℕ. Then, a sequence 𝑥=(𝑥_𝑘) of elements in 𝑋 is called lacunary ℐ^𝐹-Cauchy sequence in 𝑋 if for every 𝜀>0, 𝜆∈(0,1) and nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋, there exists 𝑚∈ℕ satisfying 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘−𝑥_𝑚∉𝒩_𝜙(𝜀,𝜆)∈ℐ. (41)

Definition 22. Let (𝑋,𝐹,∗) be RnN space, and let ℐ be an admissible ideal of ℕ. We say that a sequence 𝑥=(𝑥_𝑘) of elements in 𝑋 is called lacunary ℐ^𝐹∗-Cauchy sequence in 𝑋 if for every 𝜀>0,𝜆∈(0,1) and nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋, there exists a set 𝐾=𝑚₁<𝑚₂<⋯<𝑚_𝑘<⋯⊂ℕ (42) such that 𝐾^′={𝑟∈ℕ∶𝑚_𝑘∈𝐼_𝑟}∈ℱ(ℐ), and (𝑥_𝑚𝑘) is a lacunary Cauchy with respect to the random 𝑛-norm 𝐹.

The next theorem gives that ℐ^𝐹∗_𝜃-Cauchy sequence implies ℐ^𝐹_𝜃-Cauchy sequence.

Theorem 23. Let (𝑋,𝐹,∗) be RnN space, and let ℐ be a nontrivial ideal of ℕ. If 𝑥=(𝑥_𝑘) is a ℐ_𝜃^𝐹∗-Cauchy sequence, then 𝑥=(𝑥_𝑘) is a ℐ^𝐹_𝜃-Cauchy sequence too.

Proof. Let (𝑥_𝑘) be a ℐ_𝜃^𝐹∗-Cauchy sequence. Then, for 𝜀>0,𝜆∈(0,1) and nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋, there exist 𝐾^′∈ℱ(ℐ) and a number 𝑁∈ℕ such that 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑚𝑘−𝑥_𝑚𝑝∈𝒩_𝜙(𝜀,𝜆) (43) for every 𝑘,𝑝≥𝑁. Now, fix 𝑝=𝑚_𝑁+1. Then, for every 𝜀>0, 𝜆∈(0,1) and nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋, we have 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑚𝑘−𝑥_𝑝∈𝒩_𝜙(𝜀,𝜆)forevery𝑘≥𝑁. (44) Let 𝐻=ℕ⧵𝐾. It is obvious that 𝐻∈ℐ and 𝐴()=1𝜀,𝜆𝑟∈ℕ∶ℎ_{𝑟𝑘,𝑝∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘−𝑥_𝑝∉𝒩_𝜙()𝑚𝜀,𝜆⊂𝐻∪₁<𝑚₂<⋯<𝑚_𝑁∈ℐ. (45) Therefore, for every 𝜀>0, 𝜆∈(0,1) and nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋, we can find 𝑝∈ℕ such that 𝐴(𝜀,𝜆)∈ℐ, that is, (𝑥_𝑘) is a ℐ^𝐹_𝜃-Cauchy sequence.

Now, we will prove that ℐ^𝐹∗_𝜃-convergence implies ℐ^𝐹_𝜃-Cauchy condition in random 𝑛-normed space.

Theorem 24. Let (𝑋,𝐹,∗) be RnN space, and let ℐ be an admissible ideal of ℕ. If a sequence 𝑥=(𝑥_𝑘) is ℐ_𝜃^𝐹∗-convergent, then it is a ℐ^𝐹_𝜃-Cauchy sequence.

Proof. By assumption, there exists a set 𝐾=𝑚_𝑘∶𝑚₁<𝑚₂<⋯⊂ℕ (46) such that 𝐾^′={𝑟∈ℕ∶𝑚_𝑘∈𝐼_𝑟}∈ℱ(ℐ), and (𝑥_𝑚𝑘) is a lacunary Cauchy with respect to the random 𝑛-norm 𝐹. Choose 𝜂∈(0,1) such that (1−𝜂)∗(1−𝜂)>(1−𝜆). Since 𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1,𝑥𝑚𝑘−𝑥𝑚𝑝}(𝜀)≥𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1,𝑥𝑚𝑘−𝐿}𝜀2∗𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1,𝑥𝑚𝑝−𝐿}𝜀2>1−𝜂∗1−𝜂>1−𝜆 (47) for every 𝜀>0, 𝜆∈(0,1), each nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1 in 𝑋 and 𝑘>𝑁,𝑝>𝑁, we have (1/ℎ_𝑟)∑_{𝑘∈𝐼𝑟}(𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑚𝑘−𝑥_𝑚𝑝∉𝒩_𝐿(𝜀,𝜆)) for every 𝑘,𝑝>𝑁 and each nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋, that is, (𝑥_𝑘) in 𝑋 is an ℐ_𝜃^𝐹∗-Cauchy sequence in 𝑋. Then, by Theorem 23(𝑥_𝑘) is a ℐ^𝐹_𝜃-Cauchy sequence in RnN space.

Theorem 25. Let (𝑋,𝐹,∗) be RnN space, and let ℐ be an admissible ideal of ℕ. If a sequence 𝑥=(𝑥_𝑘) of elements in 𝑋 is ℐ^𝐹_𝜃-convergent, then it is ℐ^𝐹_𝜃-Cauchy sequence.

Proof. Suppose that (𝑥_𝑘) is ℐ^𝐹_𝜃-convergent to 𝐿∈𝑋. Let 𝜀>0,𝜆∈(0,1) and nonzero 𝑥₁,𝑥₂,…,𝑥_𝑛−1∈𝑋 be given. Then, we have 1𝐴=𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∉𝒩_𝐿𝜀2,𝜆∈ℐ. (48) This implies that 𝐴^𝑐=1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘∈𝒩_𝐿𝜀2(ℐ).,𝜆∈ℱ (49) Choose 𝜂∈(0,1) such that (1−𝜂)∗(1−𝜂)>(1−𝜆). Then, for every 𝑟∈𝐴^𝑐, 𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1,𝑥𝑘−𝑥𝑠}(𝜀)≥𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1,𝑥𝑘−𝐿}𝜀2∗𝐹_{𝑥1,𝑥2,…,𝑥𝑛−1,𝑥𝑠−𝐿}𝜀2>∗>().1−𝜂1−𝜂1−𝜆 (50) Hence{𝑟∈ℕ∶(1/ℎ_𝑟)∑_{𝑘∈𝐼𝑟}(𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘−𝑥_𝑠∈𝒩_𝜙(𝜀,𝜆))}∈ℱ(ℐ). This implies that 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝑥_𝑘−𝑥_𝑠∉𝒩_𝜙(𝜀,𝜆)∈ℐ. (51) that is, (𝑥_𝑘) is a ℐ^𝐹_𝜃-Cauchy sequence.