EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5608 ISSN 1307-5543 – ejpam.com Published by New York Business Global Necessary and Sufficient Conditions for the Equivalence of Statistical, Ideal, and Standard Convergence in G-Metric Spaces Manuharawati1,∗, Muhammad Jakfar1, Ahmad Taufik Hamzah1 1 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Negeri Surabaya, Surabaya, Indonesia Abstract. This paper investigates the conditions for the equivalence between statistical conver- gence, ideal convergence, and standard convergence in G-metric spaces. Although statistical and ideal convergence studies have been extensively developed in various settings, no prior research has explicitly explored the relationship between statistical convergence and standard convergence within G-metric spaces. By addressing this gap, we establish necessary and sufficient conditions for the equivalence of these types of convergence in G-metric spaces. Our results contribute to a deeper understanding of the interplay between these convergence notions and extend the theory of convergence in generalized metric spaces. 2020 Mathematics Subject Classifications: 40A35 Key Words and Phrases: Statistical convergence, ideal convergence, standard convergence, G-metric spaces, necessary and sufficient conditions, equivalence theorems. 1. Introduction The concept of convergence plays a central role in analysis and its applications, with various types of convergence being developed to generalize the classical notion of pointwise convergence. Among these generalizations, statistical convergence and ideal convergence have attracted considerable attention. The notion of statistical convergence, introduced by Fast [3] and further studied by Šalát [32], provides a probabilistic framework that generalizes classical convergence by considering the density of indices at which a sequence fails to converge to a limit. This approach has proven useful in various applications, from number theory to functional analysis [3, 32]. Ideal convergence, introduced by Kostyrko et al. in [20], extends statistical convergence by incorporating ideals of sets, which are collections of subsets of natural numbers closed ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5608 Email addresses: manuharawati@unesa.ac.id (Manuharawati) , muhammadjakfar@unesa.ac.id (M. Jakfar), taufikhamzahh26@gmail.com (A. Taufik Hamzah) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) Manuharawati, M.Jakfar, A. Taufik Hamzah / Eur. J. Pure Appl. Math, 18 (1) (2025), 5608 2 of 13 under certain set operations. Ideal convergence provides a flexible framework that unifies several known types of convergence, including statistical convergence and convergence with respect to filters [20]. Ideal convergence has been studied in various contexts, including Banach spaces and normed spaces, offering insights into the structure of function spaces and the behaviour of sequences [2]. Despite substantial progress in these areas, research on the relationship between statis- tical convergence and standard convergence, particularly in the setting of G-metric spaces, remains limited. G-metric spaces, introduced by Mustafa and Sims [27], generalize the notion of a metric space by defining a distance function on triplets of points rather than pairs. This structure has led to the development of various fixed-point theorems and ap- plications in nonlinear analysis [27]. Many mathematicians have conducted research on G-metric spaces. Recent results on G-metric spaces include, among others, [13], [14], [17], [18], [16], [15], [25], [26], [30], [4], [7] and [33]. However, the interaction between differ- ent types of convergence, such as statistical and standard convergence ([10], [11],[23], and [29]), in G-metric spaces has not been fully explored. The concept of convergence, particularly in G-metric spaces, has wide-ranging appli- cations across various fields. In machine learning and data analysis, these results can be applied to understand the stability of algorithms or models operating within complex metric structures, such as G-metric spaces, which better represent non-linear data relation- ships [35]. Furthermore, in optimization theory, G-metric spaces provide a framework for analyzing iterative algorithms for non-linear problems commonly encountered in dynamic programming [9]. In theoretical physics, G-metric spaces aid in modelling systems with multiple parameter interactions, making them relevant for studying dynamical systems and physical geometries [28]. Finally, in mathematical finance, this approach enhances the modelling of complex market data through statistical and ideal convergence, enabling more robust analysis of economic behaviour [16]. Thus, this study not only deepens the theoretical understanding of convergence in G-metric spaces, but also opens avenues for its application across diverse scientific disciplines. Existing research has focused mainly on the individual properties of statistical and ideal convergence in G-metric spaces [12, 31]. These studies have established important results concerning the behaviour of sequences and functions in such spaces. However, no prior study has investigated the precise relationship between statistical convergence and standard convergence in G-metric spaces, leaving a significant gap in the literature. In this paper, we aim to bridge this gap by studying the conditions under which statis- tical convergence, ideal convergence, and standard convergence are equivalent in G-metric spaces. Our main contribution is the establishment of necessary and sufficient conditions for this equivalence, which provide a comprehensive framework for understanding how these different convergence notions relate to one another. The results presented in this paper extend the theory of convergence in G-metric spaces and offer new insights into the behaviour of sequences in generalized metric structures. This paper is organized as follows. In Section 2, we provide preliminary definitions and review key results concerning the statistical and ideal convergence in G-metric spaces. Section 3 presents the main theorems, including the necessary and sufficient conditions for Manuharawati, M.Jakfar, A. Taufik Hamzah / Eur. J. Pure Appl. Math, 18 (1) (2025), 5608 3 of 13 the equivalence of statistical, ideal, and standard convergence in G-metric spaces. Finally, in Section 4, we conclude with a discussion of the implications of our results and possible directions for future research. 2. Preliminary Definition Before proceeding with the main discussion, we need to establish several definitions of the key concepts that will be explored throughout this research. In the following section, we will provide the definitions of G-metric spaces and the various types of convergence within G-metric spaces. Definition 1. [27] Let X be a non-empty set. A Function G : X ×X ×X → R+ is called a G-metric if, for all x, y, z ∈ X the following conditions are satisfied: (i) G(x, y, z) = 0 if and only if x = y = z (ii) G(x, x, y) > 0 for x ̸= y (iii) G(x, x, y) ≤ G(x, y, z), for z ̸= y (iv) G(x, y, z) = G(x, z, y) = G(y, z, x) = G(y, x, z) = G(z, x, y) = G(z, y, x) (v) G(x, y, z) ≤ G(x, a, a) +G(a, y, z) for any a ∈ X A Set X equipped with the function G is called a G-metric space and is denoted by (X,G). Definition 2. [6] Let (X,G) be a G-metric space and (xn) be a sequence in X. The sequence (xn) is said to converge to x ∈ X if lim n,m→+∞ G(x, xn, xm) = 0 means that for every ε > 0 there exists n0 ∈ N such that G(x, xn, xm) < ε for all n,m ≥ n0. x In this case, x is called the limit of the sequence (xn) denoted by xn → x or lim n→+∞ xn = x. Definition 3. [6] Let (X,G) be a G-metric space and (xn) be a sequence in X. The sequence (xn) is said to be a Cauchy sequence in the G-metric space if for every ε > 0, there exists n0 ∈ N such that for all k, n,m ≥ n0, G (xk, xn, xm) < ε. Theorem 1. [6] Let (xn) be a sequence in the G-metric space (X,G). If the sequence (xn) converges to x ∈ R, then (xn) is a Cauchy sequence. Definition 4. [1] Let (xn) be a sequence in the G-metric space (X,G). The sequence (xn) is said to converge statistically to x in the G-metric space if, for every real number ε > 0, we have lim n→+∞ ( 2 n2 |{(n1, n2) ∈ N2 : n1, n2 ≤ n,G(x, xn1 , xn2) ≥ ε}| ) = 0 and this is denoted as Gs− lim(xn) = x Manuharawati, M.Jakfar, A. Taufik Hamzah / Eur. J. Pure Appl. Math, 18 (1) (2025), 5608 4 of 13 Definition 5. [22] Let (X,G) be a G-metric space and (xn) a sequence in X. The sequence (xn) is said to be a statistical Cauchy sequence in the G-metric space if, for every ε > 0, there exists i ∈ N such that lim n→+∞ ( 2 n2 |{(n1, n2) ∈ N2 : n1, n2 ≤ n,G(xi, xn1 , xn2) ≥ ε}| ) = 0 Let I2 ⊂ 2N 2 be a nontrivial ideal on N2, where A ∈ I2 and A = {(n1, n2) : n1, n2 ∈ N)}. Definition 6. [22] Let I2 be an ideal. Let (xn) be a sequence in the G-metric space (X,G). The sequence (xn) is said to be ideally convergent to x if, for every real number ε > 0, the set {(n1, n2) ∈ N2 : G (x, xn1 , xn2) ≥ ε} ∈ I2. and this is denoted as GI − lim(xn) = x Definition 7. [22] Let (X,G) be a G-metric space and I2 an ideal. Let (xn) be a sequence in X. The sequence (xn) is said to be an ideal Cauchy sequence in the G-metric space if, for every ε > 0, there exists i ∈ N such that {(n1, n2) ∈ N2 : G(xi, xn1 , xn2) ≥ ε} ∈ I2. 3. Main Results In this section, we discuss the relationship between standard convergence, statistical convergence, and ideal convergence in the G-metric space (R, G). Theorem 2. [1] If a sequence converges to x in a G-metric space, then the sequence also statistically converges to x in the G-metric space. Proof. Let (xn) be a sequence that converges to x in a G-metric space. This means that for every real number ε > 0, there exists an index j ∈ N such that for every n,m ≥ j, we have G(x, xn, xm) < ε. If we form a set, it will take the following form: A(j) = {(n,m) ∈ N2 : n,m ≥ j,G(x, xn, xm) < ε} It is clear that because there exists j ∈ N such that for all n,m ≥ j then G(x, xn, xm) < ε. Thus, |{n : (n,m) ∈ N2, G(x, xn, xm) ≥ ε}|or|{m : (n,m) ∈ N2, G (x, xn, xm) ≥ ε}| is at most j − 1, so: lim n→+∞ ( 2 n2 |{(n,m) ∈ N2 : n,m ≥ j,G(x, xn, xm) < ε}| ) = lim n→+∞ ( 2(j − 1)n n2 ) = lim n→+∞ 2(j − 1) n ) = 0 Thus, it is proven that the sequence (xn) statistically converges to x in the G-metric space. Manuharawati, M.Jakfar, A. Taufik Hamzah / Eur. J. Pure Appl. Math, 18 (1) (2025), 5608 5 of 13 Example 1. Consider the G-metric space (R, G), where for all x, y, z ∈ R, the G-metric is defined as: G(x, y, z) = max{|x− y|+ |x− z|+ |y − z|}. The sequence ( 1 n+1 ) is statistically convergent to 0 in the G-metric space. We can investigate whether the sequence ( 1 n+1 ) also converges to 0 in the G-metric space. According to Theorem 2, the sequence ( 1 n+1 ) is statistically convergent to 0 in the G-metric space. The proof is given as follows: G(x, xn, xm) = max{|x− xn|, |x− xm|, |xn − xm|} = max {∣∣∣∣0− 1 n+ 1 ∣∣∣∣ , ∣∣∣∣0− 1 m+ 1 ∣∣∣∣ , ∣∣∣∣ 1 n+ 1 − 1 m+ 1 ∣∣∣∣} ≥ max { 1 n+ 1 , 1 m+ 1 , 1 n+ 1 − 1 m+ 1 } ≥ max { 1 n+ 1 , 1 m+ 1 } Let k be the largest integer less than or equal to 1 ε−1 , for any ε > 0, ε ∈ R. Then: lim n→+∞ ( 2 n2 |{(n1, n2) ∈ N2 : n1, n2 ≤ n,G(xi, xn1 , xn2) ≥ ε}| ) = lim n→+∞ ( 2 n2 |{(1, 1), (1, 2), (1, 3), . . . , (k, 1), (k, 2), . . . }|) ≤ lim n→+∞ ( 2nk n2 ) ≤ 2k lim n→+∞ ( 1 n ) = 2k.0 = 0 Next, we investigate whether a statistically convergent sequence is also a standard con- vergent sequence. A statistically convergent sequence in the G-metric space is not always a standard convergent sequence in the G-metric space. The following theorem provides the necessary condition for a sequence that is statistically convergent in a G-metric space to also be a standard convergent sequence in the same space. Theorem 3. Given a sequence (xn) that is statistically convergent to x in a G-metric space, if |A| = |{n : (n,m) ∈ N2, G(x, xn, xm) ≥ ε}| < +∞ or |B| = |{m : (n,m) ∈ N2, G(x, xn, xm) ≥ ε}| < +∞ then the sequence is standard convergent to x in the G- metric space. Proof. The sequence (xn) is statistically convergent to x, meaning that for every real number ε > 0, the following holds: lim n→+∞ ( 2 n2 |{(n1, n2) ∈ N2 : n1, n2 ≤ n : G(x, xn1 , xn2) ≥ ε}|) = 0 Manuharawati, M.Jakfar, A. Taufik Hamzah / Eur. J. Pure Appl. Math, 18 (1) (2025), 5608 6 of 13 If |A| = |{n : (n,m) ∈ N2, G(x, xn, xm) ≥ ε}| < +∞, then there exists a sup(A) + 1 ∈ N such that for every n,m ≥ sup(A) + 1, G(x, xn, xm) < ε. In other words, the sequence (xn) is convergent to x in the G-metric space. Similary, if |B| = |{m : (n,m) ∈ N2, G(x, xn, xm) ≥ ε}| < +∞, then there exists a sup(B) + 1 ∈ N such that for every n,m ≥ sup(B) + 1, G(x, xn, xm) < ε, and hence the sequence (xn) is standard convergent to x in the G-metric space. Example 2. Consider the G-metric space (R, G), where for all x, y, z ∈ R, the G-metric is defined as: G(x, y, z) = |x− y|+ |x− z|+ |y − z|. The sequence (xn) = ( (−1)n n ) converges to 0 in the G-metric space. It will be proven that the sequence (xn) = ( (−1)n n ) statistically converges to 0 in a G-metric space. Let j be the largest natural number less than or equal to 2 ε for every ε > 0, ε ∈ R, then lim n→+∞ ( 2 n2 |{(n1, n2) ∈ N2 : n1, n2 ≤ n,G(xi, xn1 , xn2) ≥ ε}| ) = lim n→+∞ ( 2 n2 |{(1, 1), (1, 2), (1, 3), . . . , (j, 1), (j, 2), . . . }|) ≤ lim n→+∞ ( 2nj n2 ) ≤ 2j lim n→+∞ ( 1 n ) = 2j.0 = 0 Theorem 4. Given a sequence (xn) that converges statistically to x in a G-metric space, if the sequence (xn) is monotonic, then (xn) converges ordinarily to x in the G-metric space. Proof. The sequence (xn) converging statistically to xmeans that for every real number ε > 0, the following holds: lim n→+∞ ( 2 n2 |{(n1, n2) ∈ N2 : n1, n2 ≤ n,G(x, xn1 , xn2) ≥ ε}| ) = 0 A Sequence (xn) is monotonic increasing if x1 ≤ x2 ≤ · · · ≤ xn ≤ xn+1 ≤ . . . , and mono- tonic decreasing if x1 ≥ x2 ≥ . . . xn ≥ xn+1 ≥ . . . . If for some (m,n) ∈ N2, G(x, xm, xn) ≥ ε, then for every i < n and k < m , G(x, xi, xk) ≥ ε because (xn) converges statis- tically. Additionally, since (xn) converges statistically, we have |A| = |{n : (n,m) ∈ N2, G(x, xn, xm) ≥ ε}| < +∞, let |A| = z or |B| = |{m : (n,m) ∈ N2, G(x, xn, xm) ≥ ε}| < +∞, let |B| = y. If |A| = |{n : (n,m) ∈ N2, G(x, xn, xm) ≥ ε}| < +∞, then there exists z + 1 ∈ N such that for all n,m ≥ z + 1, G(x, xn, xm) < ε, or in other words, the sequence (xn) converges ordinarily to x in the G-metric space. If Manuharawati, M.Jakfar, A. Taufik Hamzah / Eur. J. Pure Appl. Math, 18 (1) (2025), 5608 7 of 13 |B| = |{m : (n,m) ∈ N2, G(x, xn, xm) ≥ ε}| < +∞, then there exists y + 1 ∈ N such that for all n,m ≥ y+1, G(x, xn, xm) < ε, or in other words, the sequence (xn) converges ordinarily to x in the G-metric space. Example 3. Given a G-metric space, (R, G), and for every x, y, z ∈ R, the following condition holds: G (x, y, z) = max{|x− y|+ |x− z|+ |y − z|} The sequence (xn) = ( n n+1 ) converges to 1 in the G-metric space. Clearly, (xn) is an increasing sequence since n n+1 ≤ n+1 n+2 . Next, it will be proven that the sequence (xn) statistically converges to 1 in the G-metric space. Let i be the greatest integer less than or equal to 1 ε − 1 for every ε > 0, ε ∈ R, then lim n→+∞ ( 2 n2 |{(n1, n2) ∈ N2 : n1, n2 ≤ n,G(xi, xn1 , xn2) ≥ ε}| ) = lim n→+∞ ( 2 n2 |{(1, 1), (1, 2), (1, 3), . . . , (i, 1), (i, 2), . . . }|) ≤ lim n→+∞ ( 2ni n2 ) ≤ 2i lim n→+∞ ( 1 n ) = 2i.0 = 0 The statement (xn) = ( n n+1 ) is statistically convergent to 1 in the G-metric space” has been proven. Since (xn) = ( n n+1 ) is statistically convergent to 1 in the G-metric space and is a monotonic sequence, by Theorem 4, (xn) = ( n n+1 ) converges to 1 in the G-metric space. Theorem 5. [19] Given an admissible ideal I2. If a sequence converges to x in a G-metric space, then the sequence converges ideally to x in the G-metric space with the ideal I2. Proof. Let (xn) be a sequence that converges to x in the G-metric space (R, G). This means that for every real number ε > 0, there exists n0 ∈ N such that for every natural number n,m ≥ n0, we have G (x, xn, xm) < ε. If we define a set, it will form the following set: A (n0) = { (n,m) ∈ N2 : n,m ≥ n0, G (x, xn, xm) < ε } It is clear that |A (n0)| = +∞ Since there exists n0 ∈ N such that for every n,m ≥ n0, G (x, xn, xm) < ε, the number of n,m ∈ N that satisfy G (x, xn, xm) ≥ ε is at most j2, or in other words, it is finite. Thus, A (ε) ∈ I2 Therefore, the sequence (xn) is ideally convergent to x. Manuharawati, M.Jakfar, A. Taufik Hamzah / Eur. J. Pure Appl. Math, 18 (1) (2025), 5608 8 of 13 We know that the sequence (xn) = ( 1 n ) is ideally convergent to 0 in the G-metric space with one of the admissible ideals, and it also converges ordinarily to 0 in the G-metric space. Theorem 6. Let (xn) be a sequence that ideally converges to L in a G-metric space with the ideal I2. If for every A ∈ I2, |A| < lim n→+∞ ( n2 ) , then the sequence (xn) statistically converges to L in the G-metric space. Proof. It is known that the sequence (xn) is ideally convergent to L, which means that for every ε > 0, we have Aε = (n,m) ∈ N2 : G (L, xn, xm) ≥ ε ∈ I2. Since |A| < lim n→+∞ ( n2 ) for every A ∈ I2 it follows that |Aε| < +∞. Let |Aε| = lim n→+∞ (nj) , with j ∈ N, so that for every ε > 0, the following holds: lim n→+∞ ( 2 n2 |{(n,m) ∈ N2 : n,m ≥ n,G(0, xn, xm) < ε}| ) < lim n→+∞ ( nj n2 ) < lim n→+∞ ( j n ) = 0 So, the sequence (xn) statistically converges to L. Example 4. It will be proven that the sequence ( 1 n ) statistically converges to 0 in the G-metric space. It has been known that the sequence ( 1 n ) converges ideally to 0 in the G-metric space with the ideal I = A ⊂ N2 : |A| < lim n→+∞ ( n2 ) . It will also be shown that the sequence ( 1 n ) statistically converges to 0 in the G-metric space. Preliminary Analysis G (x, xn1 , xn2) = ∣∣∣∣0− 1 n1 ∣∣∣∣+ ∣∣∣∣ 1n1 − 1 n2 ∣∣∣∣+ ∣∣∣∣ 1n2 − 0 ∣∣∣∣ = 1 n1 + 1 n2 + ∣∣∣∣ 1n1 − 1 n2 ∣∣∣∣ ≥ 1 n1 + 1 n2 + (∣∣∣∣ 1n1 ∣∣∣∣− ∣∣∣∣ 1n2 ∣∣∣∣) ≥ 2 n1 To ensure 2 n1 ≥ ε, then n1 must be a natural number k such that k ≤ 2 ε . Therefore, for Manuharawati, M.Jakfar, A. Taufik Hamzah / Eur. J. Pure Appl. Math, 18 (1) (2025), 5608 9 of 13 every real number ε > 0, the following holds: lim n→+∞ ( 2 n2 |{(n1, n2) ∈ N2 : n1, n2 ≤ n,G(xi, xn1 , xn2) ≥ ε}| ) = lim n→+∞ ( 2 n2 |{(1, 1), (1, 2), (1, 3), . . . , (k, 1), (k, 2), . . . }|) ≤ lim n→+∞ ( 2nk n2 ) ≤ 2k lim n→+∞ ( 1 n ) = 2k.0 = 0 Thus, the sequence ( 1 n ) is also statistically proven to converge to 0 in the G-metric space. Theorem 7. [19] Given a sequence (xn) that statistically converges to x in the G-metric space and the ideal I2, where I2 = A : δ(A) = 0, the sequence (xn) ideally converges to x in the G-metric space. Proof. A sequence (xn) is said to statistically converge to x in a G-metric space if lim n→+∞ ( 2 n2 ∣∣(n1, n2) ∈ N2 : n1, n2 ≤ n,G (x, xn1 , xn2) ≥ ε ∣∣) = 0 Thus, the set {(n1, n2) ∈ N2 : G (x, xn1 , xn2) ≥ ε} has asymptotic density 0. This means (xn) ∈ I2, or equivalently, (xn) converges ideally to x in the G-metric space. Theorem 8. [1] If (xn) statistically converges in a G-metric space, then (xn) is a statis- tically Cauchy sequence in the G-metric space. Proof. Let ε > 0 be any real number. Suppose Gs − lim(xn) = x. Since ε is a real number and ε > 0, then ε 6 is also a real number and greater than 0. Therefore, the set {(n1, n2) ∈ N2 : G (x, xn1 , xn2) ≥ ε 6} has asymptotic density 0. Let m ∈ N be chosen such that G (x, xn1 , xm) ≥ ε 6 . Then G (xm, xn1 , xn2) ≤ G (xm, x, x) +G (x, xn1 , x) +G (x, x, xn2) ≤ 2(G (x, xn1 , xn2) +G (x, xn1 , xn2) +G (x, xn1 , xm)) < 2( ε 6 + ε 6 + ε 6 ) = ε Thus, the set {(n1, n2) ∈ N2 : G (xm, xn1 , xn2) ≥ ε} has asymptotic density 0, meaning that (xn) is an is a statistically Cauchy sequence. Theorem 9. [19] If (xn) ideally converges in a G-metric space, then (xn) is an ideal Cauchy sequence in the G-metric space. Manuharawati, M.Jakfar, A. Taufik Hamzah / Eur. J. Pure Appl. Math, 18 (1) (2025), 5608 10 of 13 Proof. Let ε > 0 be any real number. Suppose GI − lim(xn) = x. Since ε is a real number and ε > 0, then ε 6 is also a real number and greater than 0. Therefore, the set {(n1, n2) ∈ N2 : G (x, xn1 , xn2) ≥ ε 6} ∈ I. Let m ∈ N be chosen such that G (x, xn1 , xm) ≥ ε 6 . Then G (xm, xn1 , xn2) ≤ G (xm, x, x) +G (x, xn1 , x) +G (x, x, xn2) ≤ 2(G (x, xn1 , xn2) +G (x, xn1 , xn2) +G (x, xn1 , xm)) < 2( ε 6 + ε 6 + ε 6 ) = ε Thus, the set {(n1, n2) ∈ N2 : G (xm, xn1 , xn2) ≥ ε} ∈ I, which means (xn) is an ideal Cauchy sequence. Theorems 1, 2, 5, 8, and 9 yield the following corollaries: Corollary 1. If the sequence (xn) is a Cauchy sequence in a G-metric space, then the sequence (xn) is statistically Cauchy in the G-metric space. Proof. Based on Theorem 1, it can be observed that a normally convergent sequence in a G-metric space is a Cauchy sequence in the G-metric space. According to Theorem 2, if a sequence is normally convergent in a G-metric space, then it is statistically convergent in the G-metric space. Theorem 8 states that a statistically convergent sequence in a G-metric space is a statistically Cauchy sequence in the G-metric space. From these three theorems, it can be concluded that a Cauchy sequence in a G-metric space is a statistically Cauchy sequence in the G-metric space. Corollary 2. Given an admissible ideal I, if the sequence (xn) is a Cauchy sequence, then the sequence (xn) is an Ideal Cauchy sequence. Proof. According to Theorem 1, a normally convergent sequence in a G-metric space is a Cauchy sequence in the G-metric space. Based on Theorem 5, if a sequence is normally convergent in a G-metric space, then it is Ideal convergent in the G-metric space with respect to the admissible ideal. Theorem 9 states that an Ideal convergent sequence in a G-metric space is an Ideal Cauchy sequence in the G-metric space. From these three theorems, it can be concluded that a Cauchy sequence in a G-metric space. Theorem 10. Given (xn) is a sequence of real numbers with an admissible ideal I2. If (xn) is Ideal convergent to L in the G-metric space, then there exists a sequence (yn) that is statistically convergent to L in the G-metric space, such that (|xn− yn|) is Ideal convergent to 0 in the G-metric space. Proof. Let ε > 0 be a given real number. The sequence (xn) is Ideal convergent to x, which means that for each such ε, the set (n1, n2) ∈ N2 : G (x, xn1 , xn2) ≥ ε ∈ I2. The sequence (yn) is statistically convergent to x, which means that for each such ε, the following holds: lim n→+∞ ( 2 n2 ∣∣(n1, n2) ∈ N2 : n1, n2 ≤ n : G (x, yn1 , yn2) ≥ ε ∣∣) = 0 Manuharawati, M.Jakfar, A. Taufik Hamzah / Eur. J. Pure Appl. Math, 18 (1) (2025), 5608 11 of 13 Let {(n1, n2) ∈ N2 : n1, n2 ≤ n : G(x, yn1, yn2) ≥ ε} = Aε. If the sequence (yn) satisfies Aε ∈ I for each given ε, then G (x, xn1 , xn2)−G (x, yn1 , yn2) = G (0, xn1 − yn1 , xn2 − yn2) Thus, the set {(n1, n2) ∈ N2 : n1, n2 ≤ n : G ( 0, xn1 − yn1 , xn2 − yn2 ) ≥ ε} ∈ I. Therefore, the sequence (|xn − yn|) is Ideal convergent to 0. Theorem 11. Let I be a non-trivial ideal on N2. If the sequence of real numbers (xn) ideal converges to L in the metric-G space, then there exists a subsequence (xnk ) of (xn) that converges to x in the usual sense in the metric-G space, and there exists a subsequence (xmk ) that converges statistically to L in the metric-G space. Proof. Let ε > 0 be an arbitrary real number. The fact that (xn) ideal converges to L means that for any ε, the set (n,m) ∈ N2 : G (L, xn, xm) ≥ ε ∈ I. If we select a subsequence (xnk ) of (xn) such that its members satisfy G (L, xn, xm) < ε, then clearly (xnk ) converges to L. Similarly, we can choose a subsequence (xmk ) of (xn) such that lim n→+∞ ( 2 n2 |{(n1, n2) ∈ N2 : n1, n2 ≤ n,G(xi, xn1 , xn2) ≥ ε}| ) which implies that the sequence converges statistically to L. 4. Conclusion In this study, several significant theorems regarding the convergence properties of se- quences in G-metric spaces have been established. The results demonstrate a strong con- nection between different types of convergence—statistical, ideal, and standard—within the framework of G-metric spaces. Overall, this research enhances our understanding of the behaviour of sequences in G-metric spaces and the interplay between different con- vergence concepts, paving the way for further exploration in this area of metric space theory. Future research directions could include extending these results to more generalized metric spaces, such as cone G-metric spaces or fuzzy G-metric spaces, to explore whether similar equivalences and relationships hold. Another promising avenue would be to inves- tigate the applications of these convergence properties in solving fixed point problems or optimization problems, where G-metric spaces often provide a natural framework. 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