Invariant convergent and invariant ideal convergent sequence in intuitionistic fuzzy normed space

2.1 Invariant convergence in IFNS

Definition 2.1. Let (X, τ₁, τ₂, ∗ , ◊) is an IFNS and h jk = : y k + y σ ( k ) + y σ 2 ( k ) + . . . + y σ j ( k ) j + 1 for any sequence {y _n } in X. Then invariant convergence of sequence {y _n } to ξ in IFNS is define as if for all s > 0 lim j → ∞ τ 1 ( h jk - ξ , s ) = 1 and lim j → ∞ τ 2 ( h jk - ξ , s ) = 0 uniformly in k. That is for any preassigned ε ∈ (0, 1) and s > 0, there exists a j 0 = j 0 ( ε ) ∈ ℕ such that ∀j > j₀ and ∀ k ∈ ℕ , τ 1 ( h jk - ξ , s ) > 1 - ε and τ 2 ( h jk - ξ , s ) < ε . In this case, we write { y n } ⟶ σ - conv . ξ with respect to IFN (τ₁, τ₂).

Remark 2.1. In IFNS, it is obvious that usual convergence implies σ-convergence and every σ-convergent sequence is bounded sequence with respect to IFN.

Theorem 2.1. If a sequence {y _n } in IFNS is such that { y n } ⟶ σ - conv . ξ with respect to IFN (τ₁, τ₂), then limit will be unique.

Proof. Let on contrary that sequence {y _n } in IFNS is such that { y n } ⟶ σ - conv . ξ 1 as well as { y n } ⟶ σ - conv . ξ 2 with ξ₁ ≠ ξ₂ . Then we have for any preassigned ε ∈ (0, 1), there exists r ∈ (0, 1) such that (1 - r) ∗ (1 - r) ≥ (1 - ε) and r ◊ r ≤ ε. For any s > 0, there exists j 0 ′ = j 0 ′ ( r ) ∈ ℕ and j 0 ″ = j 0 ( r ) ∈ ℕ such that ∀ k ∈ ℕ , τ 1 ( h jk - ξ 1 , s / 2 ) > 1 - r and τ 2 ( h jk - ξ 1 , s / 2 ) < r , for all j > j 0 ′ as well as τ 1 ( h jk - ξ 2 , s / 2 ) > 1 - r and τ 2 ( h jk - ξ 2 , s / 2 ) < r , for all j > j 0 ″ . Now for any k ∈ ℕ and take any j > j 0 = max { j 0 ′ , j 0 ″ } , we have τ 1 ( ξ 1 - ξ 2 , s ) ≥ τ 1 ( h jk - ξ 1 , s / 2 ) ∗ τ 2 ( h jk - ξ 2 , s / 2 ) > ( 1 - r ) ∗ ( 1 - r ) ≥ ( 1 - ε ) and τ 2 ( ξ 1 - ξ 2 , s ) ≤ τ 2 ( h jk - ξ 1 , s / 2 ) ◊ τ 2 ( h jk - ξ 2 , s / 2 ) < r ◊ r ≤ ε Which implies that ξ₁ = ξ₂. Hence we reached at a contradiction.□

Theorem 2.2. If sequence {y _n } and {z _n } in IFNS is such that { y n } ⟶ σ - conv . ξ 1 and { z n } ⟶ σ - conv . ξ 2 with respect to IFN (τ₁, τ₂), then

(1) { y n + z n } ⟶ σ - conv . ξ 1 + ξ 2 with respect to IFN (τ₁, τ₂).

(2) { ay n } ⟶ σ - conv . a ξ 1 with respect to IFN (τ₁, τ₂), here a is scalar.

Proof. (1) Let sequence {y _n } and {z _n } in IFNS is such that { y n } ⟶ σ - conv . ξ 1 and { z n } ⟶ σ - conv . ξ 2 with respect to IFN (τ₁, τ₂) and h jk = : y k + y σ ( k ) + y σ 2 ( k ) + . . . + y σ j ( k ) j + 1 and h jk ′ = : z k + z σ ( k ) + z σ 2 ( k ) + . . . + z σ j ( k ) j + 1 , respectively. Then we have for any preassigned ε ∈ (0, 1), there exists r ∈ (0, 1) such that (1 - r) ∗ (1 - r) ≥ (1 - ε) and r ◊ r ≤ ε. For any s > 0, there exists j 0 ′ = j 0 ′ ( r ) ∈ ℕ and j 0 ″ = j 0 ( r ) ∈ ℕ such that ∀ k ∈ ℕ , τ 1 ( h jk - ξ 1 , s / 2 ) > 1 - r and τ 2 ( h jk - ξ 1 , s / 2 ) < r , for all j > j 0 ′ and τ 1 ( h jk ′ - ξ 2 , s / 2 ) > 1 - r and τ 2 ( h jk ′ - ξ 2 , s / 2 ) < r , for all j > j 0 ″ . Now for any k ∈ ℕ and take any j > j 0 = max { j 0 ′ , j 0 ″ } , we have τ 1 ( ( h jk + h jk ′ ) - ( ξ 1 + ξ 2 ) , s ) ≥ τ 1 ( h jk - ξ 1 , s / 2 ) ∗ τ 1 ( h jk ′ - ξ 2 , s / 2 ) > ( 1 - r ) ∗ ( 1 - r ) ≥ 1 - ε and τ 2 ( ( h jk + h jk ′ ) - ( ξ 1 + ξ 2 ) , s ) ≤ τ 2 ( h jk - ξ 1 , s / 2 ) ◊ τ 2 ( h jk ′ - ξ 2 , s / 2 ) < r ◊ r ≤ ε . Hence { y n + z n } ⟶ σ - conv . ξ 1 + ξ 2 with respect to IFN (τ₁, τ₂).

(2) If a = 0, then proof is obvious. If a ≠ 0 and { y n } ⟶ σ - conv . ξ 1 with respect to IFN (τ₁, τ₂) then for any preassigned ε ∈ (0, 1) and s > 0, ∃ j 0 = j 0 ( ε ) ∈ ℕ such that for all j > j₀ and for all k ∈ ℕ τ 1 ( h jk - ξ 1 , s / | a | ) > 1 - ε and τ 2 ( h jk - ξ 1 , s / | a | ) < ε , for all j > j 0 Now for any j > j₀ and for any k ∈ ℕ , we have τ 1 ( ah jk - a ξ 1 , s ) = τ 1 ( h jk - ξ 1 , s / | a | ) > 1 - ε and τ 1 ( ah jk - a ξ 1 , s ) = τ 1 ( h jk - ξ 1 , s / | a | ) < ε . Hence { ay n } ⟶ σ - conv . a ξ 1 with respect to IFN (τ₁, τ₂).□

Definition 2.2. Let (X, τ₁, τ₂, ∗ , ◊) is an IFNS and h jk = : y k + y σ ( k ) + y σ 2 ( k ) + . . . + y σ j ( k ) j + 1 for any sequence {y _n } in X. Then invariant Cauchy criterion of sequence {y _n } in IFNS is define as if for any preassigned ε ∈ (0, 1) and s > 0, there exists a j 0 = j 0 ( ε ) ∈ ℕ such that ∀j, j′ > j₀ and ∀ k , k ′ ∈ ℕ , τ 1 ( h jk - h j ′ k ′ , s ) > 1 - ε and τ 2 ( h jk - h j ′ k ′ , s ) < ε .(2.1)

Theorem 2.3. A sequence in IFNS is invariant convergent if and only if sequence is invariant Cauchy with respect to IFN (τ₁, τ₂).

Proof. Let sequence {y _n } is invariant convergent to ξ in an IFNS and h jk = : y k + y σ ( k ) + y σ 2 ( k ) + . . . + y σ j ( k ) j + 1 . Then for any preassigned ε ∈ (0, 1), there exists r ∈ (0, 1) such that (1 - r) ∗ (1 - r) ≥ (1 - ε) and r ◊ r ≤ ε. For any s > 0, there exists j 0 = j 0 ( r ) ∈ ℕ such that ∀ k ∈ ℕ , τ 1 ( h jk - ξ , s / 2 ) > 1 - r and τ 2 ( h jk - ξ , s / 2 ) < r , for all j > j 0 . It is obvious that ∀j, j′ > j₀ and ∀ k , k ′ ∈ ℕ , we have τ 1 ( h jk - h j ′ k ′ , s ) ≥ τ 1 ( h jk - ξ , s / 2 ) ∗ τ 1 ( h j ′ k ′ - ξ , s / 2 ) > ( 1 - r ) ∗ ( 1 - r ) ≥ 1 - ε and τ 2 ( h jk - h j ′ k ′ , s ) ≤ τ 2 ( h jk - ξ , s / 2 ) ◊ τ 2 ( h j ′ k ′ - ξ , s / 2 ) < r ◊ r ≤ ε . Hence sequence {y _n } is invariant Cauchy with respect to IFN (τ₁, τ₂).

Conversely, let sequence {y _n } is invariant Cauchy with respect to IFN (τ₁, τ₂). Then for any preassigned ε ∈ (0, 1) there exists r ∈ (0, 1) such that (1 - r) ∗ (1 - r) ≥ (1 - ε) and r ◊ r ≤ ε. For any s > 0, there exists a j 0 = j 0 ( ε ) ∈ ℕ such that ∀j, j′ > j₀ and ∀ k , k ′ ∈ ℕ , τ 1 ( h jk - h j ′ k ′ , s / 2 ) > 1 - r and τ 2 ( h jk - h j ′ k ′ , s / 2 ) < r . Fixing k = k₀, hence (2.1) cauchy implies considering k = k′ = k₀ that { h jk 0 } j = 0 ∞ satisfies Cauchy criterion so it converges to some ξ. For any s > 0, there exists j 1 = j 1 ( r ) ∈ ℕ such that τ 1 ( h jk 0 - ξ , s / 2 ) > 1 - r and τ 2 ( h jk 0 - ξ , s / 2 ) < r , for all j > j 1 . Now for any k ∈ ℕ and take any j > j′ = max {j₀, j₁}, we have τ 1 ( h jk - ξ , s ) ≥ τ 1 ( h jk - h jk 0 , s / 2 ) ∗ τ 1 ( h jk 0 - ξ , s / 2 ) > ( 1 - r ) ∗ ( 1 - r ) ≥ 1 - ε and τ 2 ( h jk - ξ , s ) ≤ τ 2 ( h jk - h jk 0 , s / 2 ) ◊ τ 2 ( h jk 0 - ξ , s / 2 ) < r ◊ r ≤ ε . Hence { y n } ⟶ σ - conv . ξ with respect to IFN (τ₁, τ₂).□

Definition 2.3. Let (X, τ₁, τ₂, ∗ , ◊) is an IFNS and I _σ is ideal of those subsets of ℕ whose σ-density is zero. Then invariant ideal convergence(or σ-ideal convergence) of a sequence {y _n } to ξ in IFNS is define as if for any preassigned ε ∈ (0, 1) and s > 0 the following set A ε = { n ∈ ℕ : τ 1 ( y n - ξ , s ) ≤ 1 - ε or τ 2 ( y n - ξ , s ) ≥ ε } ∈ I σ , means σ-density of set A _ε is zero. That is for all ε ∈ (0, 1) and s > 0 lim j → ∞ 1 j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 1 ( y n - ξ , s ) ≤ 1 - ε or τ 2 ( y n - ξ , s ) ≥ ε } | = 0 uniformly in k . In this case, we write { y n } ⟶ σ - I - cgt . ξ with respect to IFN (τ₁, τ₂).

Definition 2.4. Let (X, τ₁, τ₂, ∗ , ◊) is an IFNS, I is a non-trivial admissible ideal of ℕ and h jk = : y k + y σ ( k ) + y σ 2 ( k ) + . . . + y σ j ( k ) j + 1 for any sequence {y _n } in X. Then ideal invariant convergence(or ideal σ-convergence) of a sequence {y _n } to ξ in IFNS is define as if for any preassigned ε ∈ (0, 1), s > 0 and for all k ∈ ℕ , { j ∈ ℕ : τ 1 ( h jk - ξ , s ) ≤ 1 - ε or τ 2 ( h jk - ξ , s ) ≥ ε } ∈ I . In this case, we write { y n } ⟶ I - σ - cgt . ξ with respect to IFN (τ₁, τ₂).

Remark 2.2. In IFNS,

1) clearly, invariant I-convergence implies I-convergence implies I-invariant convergence but converse need not to be true.

2) if a sequence {y _n } is invariant convergent to ξ then obviously sequence will be ideal invariant convergent to ξ but not invariant ideal convergent in general.

Example 2.1. In Example 1.1, suppose X is set of real numbers with usual norm and I = I _δ. Let sequence {y _n } is such that y n = { 1 , if n is prime number 0 , otherwise . Clearly, sequence {y _n } is invariant I _δ-convergent so it is I _δ-convergent and I _δ-invariant convergent but unfortunately sequence is not convergent.

Example 2.2. In Example 1.1, suppose X is set of real numbers with usual norm and I = I _δ. Let σ (k) = k + 1 and sequence {y _n } is such that y n = { 1 , if n ∈ P 0 , if n ∉ P . Here P is a set as defined in Example 1.2. Clearly, sequence {y _n } is I _δ-convergent but not invariant I _δ-convergent.

Example 2.3. In Example 1.1, suppose X is set of real numbers with usual norm and I = I _δ. Let σ (k) = k + 1 and sequence {y _n } is such that y n = { 1 , if n is not divisible by 2 0 , if n is divisible by 2 . Clearly, sequence {y _n } is invariant convergent to 1/2 as well as I _δ-invariant convergent to 1/2 but unfortunately sequence is neither I _δ-convergent nor invariant I _δ-convergent.

Parallel results as Theorem 2.1, 2.2 can be easily establised for σ-ideal convergence and ideal σ-convergence with respect to IFN (τ₁, τ₂).

Theorem 2.4. Let H σ ( τ 1 , τ 2 ) ( I ) be the space of all bounded and invariant ideal convergent sequences with respect to IFN (τ₁, τ₂), then H σ ( τ 1 , τ 2 ) ( I ) and is a closed linear subset of , here denotes the space of all bounded sequences with respect to IFN (τ₁, τ₂).

Proof. Let sequence { y m } m ∈ ℕ ⊆ H σ ( τ 1 , τ 2 ) ( I ) is convergent to y such that . Now we show that y ∈ H σ ( τ 1 , τ 2 ) ( I ) . Let sequence y ^m is invariant ideal convergent to ξ _m with respact to IFN (τ₁, τ₂), for each m ∈ ℕ and take sequence {ε _m } with ε m = ε 2 m , for preassigned ε > 0. Clearly, we can choose a natural number m such that | | y m - y | | ∞ = sup m { τ 2 ( y - y m , s ) } < ε m 4 . For any given δ ∈ (0, 1), there exists a natural number j₀ such that for all j ≥ j₀ 1 j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 1 ( y n m − ξ m , s ) ≤ 1 − ( ϵ m / 4 ) or τ 2 ( y n m − ξ m , s ) ≥ ϵ m / 4 } | < δ 3 < δ < 1 and 1 j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 1 ( y n m + 1 − ξ m + 1 , s ) ≤ 1 − ( ϵ m + 1 / 4 ) or τ 2 ( y n m + 1 − ξ m + 1 , s ) ≥ ϵ m + 1 / 4 } | < δ 3 < δ < 1 Therefore, ∃ a natural number n such that σ (k) ≤ n ≤ σ ^j  (k) for which τ 1 ( y n m − ξ m , s ) > 1 − ( ϵ m / 4 ) or τ 2 ( y n m − ξ m , s ) < ϵ m / 4 and τ 1 ( y n m + 1 − ξ m + 1 , s ) > 1 − ( ϵ m + 1 / 4 ) or τ 2 ( y n m + 1 − ξ m + 1 , s ) < ϵ m + 1 / 4 . Hence τ 2 ( ξ m − ξ m + 1 , s ) ≤ τ 2 ( ξ m − y n m , s / 3 ) ⋄ τ 2 ( y n m − y n m + 1 , s / 3 ) ⋄ τ 2 ( ξ m + 1 − y n m + 1 , s / 3 ) ≤ τ 2 ( ξ m − y n m , s / 3 ) ⋄ τ 2 ( ξ m + 1 − y n m + 1 , s / 3 ) ⋄ sup m { τ 2 ( y − y m + 1 , s ) } ⋄ sup m { τ 2 ( y − y m , s ) } < ϵ m 4 ⋄ ϵ m + 1 4 ⋄ ϵ m 4 ⋄ ϵ m + 1 4 ≤ ϵ m . Hence sequence { ξ m } m ∈ ℕ satisfies Cauchy criterion with respect to IFN (τ₁, τ₂). Therefore, ∃ a ξ such that {ξ _m } converges to ξ. Now we show that y invariant ideal convergent to ξ with respect to (τ₁, τ₂). For any preassigned ε ∈ (0, 1) and s > 0, take m ∈ ℕ such that ε m < ε 4 , | | y m - y | | ∞ = sup m { τ 2 ( y - y m , s ) } < ε 4 , τ 1 ( ξ - ξ m , s ) > 1 - ε 4 or τ 2 ( ξ - ξ m , s ) < ε 4 . Hence we have

1 j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 2 ( y n - ξ , s ) ≥ ε } | ≤ 1 j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 2 ( y n m - ξ m , s ) ◊ sup m { τ 2 ( y n - y n m , s ) } ◊ τ 2 ( ξ m - ξ , s ) ≥ ε } |(2.2)

≤ 1 j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 2 ( y n m - ξ m , s ) ≥ ε / 2 } | .(2.4) Therefore,

1 j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 1 ( y n - ξ , s ) ≤ 1 - ε or τ 2 ( y n - ξ , s ) ≥ ε } |

≤ 1 j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 1 ( y n m - ξ m , s ) ≤ 1 - ε / 2 or τ 2 ( y n m - ξ m , s ) ≥ ε / 2 } | → 0 as j → ∞ .(2.5) Hence y = {y _n } is invariant ideal convergent to ξ with respect to (τ₁, τ₂).□

A similar result as Theorem 2.4 can be proved for ideal invariant convergence.

Proposition 2.5. Let (X, τ₁, τ₂, ∗ , ◊) is an IFNS. Let {y _n } be a sequence in X. For every ε > 0 and s > 0, these following are equivalent:

(a) { y n } ⟶ σ - I - cgt . ξ with respect to IFN (τ₁, τ₂);

(b) { n ∈ ℕ : τ 1 ¯ ( y n - ξ , s ) ≥ ε } ∈ I σ and { n ∈ ℕ : τ 2 ( y n - ξ , s ) ≥ ε } ∈ I σ ;

(c) { n ∈ ℕ : τ 1 ¯ ( y n - ξ , s ) < ε and τ 2 ( y n - ξ , s ) < ε } ∈ F ( I σ ) ;

(d) { n ∈ ℕ : τ 1 ¯ ( y n - ξ , s ) < ε } ∈ F ( I σ ) and { n ∈ ℕ : τ 2 ( y n - ξ , s ) < ε } ∈ F ( I σ ) ;

(e) I σ - lim τ 1 ¯ ( y n - ξ , s ) = 0 and I _σ - lim τ₂ (y _n  - ξ, s) =0 .

Proposition 2.6. Let (X, τ₁, τ₂, ∗ , ◊) is an IFNS, I be a non-trivial admissible ideal of ℕ and h jk = : y k + y σ ( k ) + y σ 2 ( k ) + . . . + y σ j ( k ) j + 1 for any sequence {y _n } in X. For every ε > 0, s > 0 and for all k ∈ ℕ , these following are equivalent:

(a) { y n } ⟶ I - σ - cgt . ξ with respect to IFN (τ₁, τ₂);

(b) { j ∈ ℕ : τ 1 ¯ ( h jk - ξ , s ) ≥ ε } ∈ I and { j ∈ ℕ : τ 2 ( h jk - ξ , s ) ≥ ε } ∈ I ;

(c) { j ∈ ℕ : τ 1 ¯ ( h jk - ξ , s ) < ε and τ 2 ( h jk - ξ , s ) < ε } ∈ F ( I ) ;

(d) { j ∈ ℕ : τ 1 ¯ ( h jk - ξ , s ) < ε } ∈ F ( I ) and { j ∈ ℕ : τ 2 ( h jk - ξ , s ) < ε } ∈ F ( I ) ;

(e) I - lim τ 1 ¯ ( h jk - ξ , s ) = 0 and I - lim τ₂ (h _jk  - ξ, s) =0 .

Theorem 2.7. Let (X, τ₁, τ₂, ∗ , ◊) is an IFNS, then every bounded invariant ideal convergent sequence in X is ideal invariant convergent with respect to IFN (τ₁, τ₂).

Proof. Let sequence {y _n } is bounded and invariant ideal convergent to ξ in X and h jk = : y k + y σ ( k ) + y σ 2 ( k ) + . . . + y σ j ( k ) j + 1 Suppose for any ε ∈ (0, 1), P ( σ , ε ) = { σ ( k ) ≤ n ≤ σ j ( k ) : τ 1 ¯ ( y n - ξ , s ) ≥ ε or τ 2 ( y n - ξ , s ) ≥ ε } . Then τ 1 ¯ ( h jk - ξ , s ) = τ 1 ¯ ( 1 j ∑ i = 1 j y σ i ( k ) - ξ , s ) = τ 1 ¯ ( ∑ n = σ ( k ) σ j ( k ) 1 j ( y n - ξ ) , s ) ≤ max { τ 1 ¯ ( 1 j ( y n - ξ ) , s j ) : σ ( k ) ≤ n ≤ σ j ( k ) } ≤ max { τ 1 ¯ ( 1 j ( y n - ξ ) , s j ) : n ∈ P ( σ , ε ) } K ( σ ( k ) , σ j ( k ) ) ≤ 1 j max n { τ 1 ¯ ( y n - ξ , s / j ) } max k { K ( σ ( k ) , σ j ( k ) ) } = 1 j max n { τ 1 ¯ ( y n - ξ , s / j ) } D j → 0 as j → ∞ . and τ 2 ( h jk - ξ , s ) = τ 2 ( 1 j ∑ i = 1 j y σ i ( k ) - ξ , s ) = τ 2 ( ∑ n = σ ( k ) σ j ( k ) 1 j ( y n - ξ ) , s ) ≤ max { τ 2 ( 1 j ( y n - ξ ) , s j ) : σ ( k ) ≤ n ≤ σ j ( k ) } ≤ max { τ 2 ( 1 j ( y n - ξ ) , s j ) : n ∈ P ( σ , ε ) } K ( σ ( k ) , σ j ( k ) ) ≤ 1 j max n { τ 2 ( y n - ξ , s / j ) } max k { K ( σ ( k ) , σ j ( k ) ) } = 1 j max n { τ 2 ( y n - ξ , s / j ) } D j → 0 as j → ∞ . Hence sequence {y _n } is invariant convergent to ξ and hence it is ideal invariant convergent.□

Example 2.4. Take IFNS ( ℝ , τ 1 , τ 2 , ∗ , ◊ ) with τ₁ and τ₂ as Example 1.1 with usual norm and I = I _δ. Let σ (k) = k + 1 and sequence {y _n } is such that y n = { 1 , if n is not divisible by 2 - 1 , if n is divisible by 2 . Clearly, sequence {y _n } is invariant convergent to zero so I _δ-invariant convergent to zero but unfortunately sequence is not invariant I _δ-convergent.

Definition 2.5. Let (X, τ₁, τ₂, ∗ , ◊) is an IFNS, then strongly invariant convergence(or strongly σ-convergence) of sequence {y _n } to ξ in IFNS is define as if for all s > 0 lim j → ∞ 1 j ∑ i = 1 j τ 1 ( y σ i ( k ) - ξ , s ) = 1 and lim j → ∞ 1 j ∑ i = 1 j τ 2 ( y σ i ( k ) - ξ , s ) = 0 uniformly in k. In this case, we write { y n } ⟶ st - σ - cgt . ξ with respect to IFN (τ₁, τ₂).

Theorem 2.8. Let {y _n } be a sequence in IFNS such that {y _n } is strongly σ-convergent sequence then {y _n } is σ-ideal convergent with respect to IFN.

Proof. Let (X, τ₁, τ₂, ∗ , ◊) is an IFNS and {y _n } is a sequence in X such that {y _n } is strongly σ-convergent to ξ with respect to IFN (τ₁, τ₂). Then for any preassigned ε ∈ (0, 1) and s > 0, there exists a j 0 = j 0 ( ε ) ∈ ℕ such that ∀j > j₀ and ∀ k ∈ ℕ ,

1 j ∑ i = 1 j τ 1 ¯ ( y σ i ( k ) - ξ , s ) < ε and 1 j ∑ i = 1 j τ 2 ( y σ i ( k ) - ξ , s ) < ε . From the following inequalities | 1 j ( ∑ i = 1 j τ 1 ¯ ( y σ i ( k ) - ξ , s ) ) | ≥ ε j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 1 ¯ ( y n - ξ , s ) ≥ ε } | and 1 j ∑ i = 1 j τ 2 ( y σ i ( k ) - ξ , s ) ≥ ε j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 2 ( y n - ξ , s ) ≥ ε } | , we have for any δ > 0 A = { j ∈ ℕ : 1 j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 1 ¯ ( y n - ξ , s ) ≥ ε } | ≥ δ } ⊂ { 1 , 2 , . . . , j 0 } and B = { j ∈ ℕ : 1 j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 2 ( y n - ξ , s ) ≥ ε } | ≥ δ } ⊂ { 1 , 2 , . . . , j 0 } So we have A ∪ B ⊂ {1, 2, . . . , j₀}. Therefore { n ∈ ℕ : τ 1 ¯ ( y n - ξ , s ) ≥ ε or τ 2 ( y n - ξ , s ) ≥ ε } ∈ I σ . Hence { y n } ⟶ σ - I - cgt . ξ with respect to IFN (τ₁, τ₂).□

Theorem 2.9. Let {y _n } be a sequence in IFNS such that {y _n } is bounded and σ-ideal convergent sequence then {y _n } is strongly σ-convergent with respect to IFN.

Proof. Let (X, τ₁, τ₂, ∗ , ◊) is an IFNS and {y _n } is a sequence in X such that {y _n } is bounded and σ-ideal convergent to ξ with respect to IFN (τ₁, τ₂). Then for any preassigned ε, δ ∈ (0, 1), s > 0 and for all k ∈ ℕ , lim j → ∞ 1 j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 1 ¯ ( y n - ξ , s ) ≥ ε or τ 2 ( y n - ξ , s ) ≥ ε } | = 0 uniformly in k(2.6) and ∃s > 0 and M ∈ (0, 1) such that τ 1 ¯ ( y n - ξ , s ) < M and τ 2 ( y n - ξ , s ) < M for all n . Now since for each k ∈ ℕ | 1 j ( ∑ i = 1 j τ 1 ¯ ( y σ i ( k ) − ξ , s ) ) | = | 1 j ( ∑ n = σ ( k ) , τ 1 ¯ ( y n − ξ , s ) ≥ ϵ σ j ( k ) τ 1 ¯ ( y n − ξ , s ) + ∑ n = σ ( k ) , τ 1 ¯ ( y n − ξ , s ) < ϵ σ j ( k ) τ 1 ¯ ( y n − ξ , s ) ) | ≤ | 1 j ( ∑ n = σ ( k ) , τ 1 ¯ ( y n − ξ , s ) ≥ ϵ σ j ( k ) τ 1 ¯ ( y n − ξ , s ) ) | + | 1 j ( ∑ n = σ ( k ) , τ 1 ¯ ( y n − ξ , s ) < ϵ σ j ( k ) τ 1 ¯ ( y n − ξ , s ) ) | < M j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 1 ¯ ( y n − ξ , s ) ≥ ϵ } | + ϵ and | 1 j ( ∑ i = 1 j τ 2 ( y σ i ( k ) − ξ , s ) ) | = | 1 j ( ∑ n = σ ( k ) , τ 2 ( y n − ξ , s ) ≥ ϵ σ j ( k ) τ 2 ( y n − ξ , s ) + ∑ n = σ ( k ) , τ 2 ( y n − ξ , s ) < ϵ σ j ( k ) τ 2 ( y n − ξ , s ) ) | ≤ | 1 j ( ∑ n = σ ( k ) , τ 2 ( y n − ξ , s ) ≥ ϵ σ j ( k ) τ 2 ( y n − ξ , s ) ) | + | 1 j ( ∑ n = σ ( k ) , τ 2 ( y n − ξ , s ) < ϵ σ j ( k ) τ 2 ( y n − ξ , s ) ) | < M j | { σ ( k ) ≤ n ≤ σ j ( k ) : τ 2 ( y n − ξ , s ) ≥ ϵ } | + ϵ . Hence from (2.6), we can conclude that sequence {y _n } is strongly σ-convergent to ξ in X.□