On Tauberian theorems for Cesàro summability of sequences of fuzzy numbers

Abstract

In this paper, we show under which conditions the limit of the distance between nth term of a sequence of fuzzy numbers and nth term of its Cesàro mean of order one tends to zero. As corollaries we prove several Tauberian theorems for Cesàro summability of sequences of fuzzy numbers.

Keywords

Fuzzy numbers sequences of fuzzy numbers Cesàro summability

1 Introduction

Let D denote the set of all closed and bounded intervals $A = [\underline{A}, \bar{A}]$ on the real line $ℝ$ .

For A, B ∈ D, we define $d (A, B) = max (| \underline{A}, \underline{B} |, | \bar{A}, \bar{B} |)$ where $A = [\underline{A}, \bar{A}]$ and $B = [\underline{B}, \bar{B}]$ . It is known that (D, d) is a complete metric space.

A function $u : ℝ \to [0, 1]$ which is normal and fuzzy convex is called a fuzzy number. Let E¹ denote the set of all fuzzy numbers which are upper semi-continuous and have a compact support.

For 0 < α ≤ 1, the α-level set of a fuzzy number u denoted by u^α is defined by $u^{α} = {t \in ℝ : u (t) \geq α} .$

The set u⁰ is defined as the closure of the set ${t \in ℝ : u (t) > 0}$ .

We define $\bar{d} : E^{1} \times E^{1} \to ℝ_{+} \cup {0}$ by $\bar{d} (u, v) = sup_{0 \leq α \leq 1} d (u^{α}, v^{α}) .$

The following properties [4] will be needed in the sequel.

$\bar{d} (u + w, v + w) = \bar{d} (u, v)$ for all u, v, w ∈ E¹

$\bar{d} (ku, kv) = | k | \bar{d} (u, v)$ for all $u, v \in E^{1}, k \in ℝ$

$\bar{d} (u + v, w + z) \leq \bar{d} (u, w) + \bar{d} (v, z)$ for all u, v, w, z ∈ E¹

$| \bar{d} (u, 0) - \bar{d} (v, 0) | \leq \bar{d} (u, v) \leq \bar{d} (u, 0) + \bar{d} (v, 0)$ for all u, v ∈ E¹

A sequence u = (u_n) of fuzzy numbers is a function u from the set

ℕ \cup {0}

of all positive integer into E¹.

A sequence (u_n) of fuzzy numbers is said to be convergent to L, written as $lim_{n \to \infty} u_{n} = L$ , if for every ɛ > 0 there exists a positive integer N₀ such that $\bar{d} (u_{n}, L) < ɛ$ for n > N₀.

The arithmetic means σ_n of (u_n) are defined by $σ_{n} = \frac{1}{n + 1} \sum_{j = 0}^{n} u_{j}$ for all $n \in ℕ \cup {0}$ . A sequence (u_n) of fuzzy numbers is said to be Cesàro summable to L if $lim_{n \to \infty} σ_{n} = L .$ (1) If the limit $lim_{n \to \infty} u_{n} = L$ (2) exists, then the limit (1) also exists ([16]).

Subrahmanyam [16] showed by an example of a sequence of fuzzy numbers that the converse of this implication was not held. The existence of the limit (1) with some suitable condition may imply (2). Such a condition is called a Tauberian condition and the resulting theorem is called a Tauberian theorem.

A fuzzy analogue of a classical two-sided Tauberian theorem of Hardy [12] for Cesàro summability method is given by Subrahmanyam [16] and it states that if (1) is satisfied and there exists a positive constant C such that $n \bar{d} (u_{n}, u_{n - 1}) \leq C (u_{- 1} = 0) (n \in ℕ \cup {0})$ (3) then (u_n) is convergent to L.

Fuzzy analogue of the concept of slowly oscillating sequence is introduced by Subrahmanyam [16].

A sequence (u_n) of fuzzy numbers is said to be slowly oscillating if $\bar{d} (u_{m}, u_{n}) = o (1), 1 \leq \frac{m}{n} \to 1, m, n \to \infty .$ (4) Note that the slow oscillation is introduced by Schmidt [14] for sequences of real numbers. It is clear that (3) implies (3).

It is also proved by Subrahmanyam [16] that if (1) is satisfied and (u_n) is slowly oscillating, then (u_n) is convergent to L. Talo and Çakan [18] have given necessary and sufficient Tauberian conditions, under which convergence follows from Cesàro convergence of sequences of fuzzy numbers. Altın et al. [2] have studied the concept of statistical summability (C, 1) for sequences of fuzzy real numbers and obtained a Tauberian theorem. Recently, Çanak [5 –7] have established some Tauberian theorems for Cesàro summability of sequences of fuzzy numbers. Furthermore, Çanak [8] and Önder et al. [13] have proved some Tauberian type theorems for the weighted mean method of summability of sequences of fuzzy numbers. In addition, sequences of fuzzy numbers have been discussed by Altınok and Mursaleen [3], Çolak et al. [9], Alotaibi et al. [1] and many others.

The fuzzy analogue of the concept of slow oscillation defined by Schmidt may not be operational to handle in some cases. For this reason we use the following fuzzy analogue of the concept of slow oscillation given in [15] and used in [10, 11].

A sequence (u_n) of fuzzy numbers is slowly oscillating if $lim_{λ \to 1^{+}} \underset{n \to \infty}{lim sup} max_{n + 1 \leq k \leq [λ n]} \bar{d} (u_{k}, u_{n}) = 0 .$ (5)

Here, [λn] denotes the integer part of λn. An equivalent reformulation of (5) is the following:

For every ɛ > 0 there exist n₁ > 0 and λ > 1 such that $\bar{d} (u_{k}, u_{n}) \leq ɛ$ whenever n₁ < n < k ≤ [λn].

In this paper we show that the limit of the distance between nth term of a sequence of fuzzy numbers and nth term of its Cesàro mean of order is zero under the conditions (12) and (13) (resp. (14) and (15)). As corollaries we prove several Tauberian theorems for Cesàro summability of sequences of fuzzy numbers.

2 Lemmas

For the proof of Theorems 3.1 and 3.2, we need the following Lemmas.

Lemma 2.1. [17] Let (u_n) be a sequence of fuzzy numbers. Then,

(i) For λ > 1 and sufficiently large n, $\frac{[λ n] + 1}{[λ n] - n} σ_{[λ n]} + σ_{n} = \frac{[λ n] + 1}{[λ n] - n} σ_{n} + τ_{n, [λ n]}^{>} .$ (6) (ii) For 0 < λ < 1 and sufficiently large n, $\frac{[λ n] + 1}{n - [λ n]} σ_{[λ n]} + τ_{n, [λ n]}^{<} = (\frac{[λ n] + 1}{n - [λ n]} + 1) σ_{n} .$ (7)

Lemma 2.2.Let (u_n) be a sequence of fuzzy numbers. Then, (i) For λ > 1 and sufficiently large n, $\bar{d} (τ_{n, [λ n]}^{>}, σ_{[λ n]}) \leq \frac{2 [λ n] - n + 1}{[λ n] - n} \bar{d} (σ_{n}, σ_{[λ n]}) .$ (8) (ii) For 0 < λ < 1 and sufficiently large n, $\bar{d} (τ_{n, [λ n]}^{<}, σ_{[λ n]}) \leq \frac{n + 1}{n - [λ n]} \bar{d} (σ_{n}, σ_{[λ n]}) .$ (9)

Proof. (i) For λ > 1 and sufficiently large n, we have $\begin{matrix} \frac{[λ n] - n}{[λ n] + 1} \bar{d} (τ_{n, [λ n]}^{>}, σ_{[λ n]}) \\ = \bar{d} (\frac{[λ n] - n}{[λ n] + 1} τ_{n, [λ n]}^{>}, \frac{[λ n] - n}{[λ n] + 1} σ_{[λ n]}) \\ = \bar{d} (\frac{[λ n] - n}{[λ n] + 1} τ_{n, [λ n]}^{>} + σ_{n}, \frac{[λ n] - n}{[λ n] + 1} σ_{[λ n]} + σ_{n}) \\ = \bar{d} (σ_{[λ n]} + \frac{[λ n] - n}{[λ n] + 1} σ_{n}, \frac{[λ n] - n}{[λ n] + 1} σ_{[λ n]} + σ_{n}) \\ \leq \bar{d} (σ_{[λ n]}, σ_{n}) + \frac{[λ n] - n}{[λ n] + 1} \bar{d} (σ_{[λ n]}, σ_{n}) \\ = \frac{2 [λ n] - n + 1}{[λ n] + 1} \bar{d} (σ_{[λ n]}, σ_{n}), \end{matrix}$ which is equivalent to (8).

(ii) For 0 < λ < 1 and sufficiently large n, we have $\begin{matrix} \frac{n - [λ n]}{[λ n] + 1} \bar{d} (τ_{n, [λ n]}^{<}, σ_{[λ n]}) \\ = \bar{d} (\frac{n - [λ n]}{[λ n] + 1} τ_{n, [λ n]}^{<}, \frac{n - [λ n]}{[λ n] + 1} σ_{[λ n]}) \\ = \bar{d} (\frac{n - [λ n]}{[λ n] + 1} τ_{n, [λ n]}^{>} + σ_{[λ n]}, \frac{n - [λ n]}{[λ n] + 1} σ_{[λ n]} + σ_{[λ n]}) \\ = \bar{d} (σ_{n} + \frac{n - [λ n]}{[λ n] + 1} σ_{n}, \frac{n - [λ n]}{[λ n] + 1} σ_{[λ n]} + σ_{[λ n]}) \\ \leq \bar{d} (σ_{[λ n]}, σ_{n}) + \frac{n - [λ n]}{[λ n] + 1} \bar{d} (σ_{[λ n]}, σ_{n}) \\ = \frac{n + 1}{[λ n] + 1} \bar{d} (σ_{[λ n]}, σ_{n}), \end{matrix}$ which is equivalent to (9). □

Lemma 2.3.Let (u_n) be a sequence of fuzzy numbers. Then, (i) For λ > 1 and sufficiently large n,

$\begin{matrix} \bar{d} (u_{n}, σ_{n}) & \leq & \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) \\ + \frac{3 [λ n] - 2 n + 1}{[λ n] - n} \bar{d} (σ_{n}, σ_{[λ n]}) . \end{matrix}$ (10) (ii) For 0 < λ < 1 and sufficiently large n,

$\begin{matrix} \bar{d} (u_{n}, σ_{n}) & \leq & \bar{d} (u_{n}, τ_{n, [λ n]}^{<}) \\ + \frac{2 n - [λ n] + 1}{n - [λ n]} \bar{d} (σ_{n}, σ_{[λ n]}) . \end{matrix}$ (11)

Proof. (i) We have $\begin{matrix} \bar{d} (u_{n}, σ_{n}) \\ = \bar{d} (u_{n} + τ_{n, [λ n]}^{>}, σ_{n} + τ_{n, [λ n]}^{>}) \\ \leq \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) + \bar{d} (τ_{n, [λ n]}^{>}, σ_{n}) \\ = \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) + \bar{d} (τ_{n, [λ n]}^{>} + σ_{[λ n]}, σ_{n} + σ_{[λ n]}) \\ \leq \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) + \bar{d} (τ_{n, [λ n]}^{>}, σ_{[λ n]}) + \bar{d} (σ_{[λ n]}, σ_{n}) . \end{matrix}$ It follows by Lemma 2.2 (i) that $\begin{matrix} \bar{d} (u_{n}, σ_{n}) & \leq & \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) \\ + \frac{3 [λ n] - 2 n + 1}{[λ n] - n} \bar{d} (σ_{n}, σ_{[λ n]}) . \end{matrix}$ (ii) We have $\begin{matrix} \bar{d} (u_{n}, σ_{n}) \\ = \bar{d} (u_{n} + τ_{n, [λ n]}^{<}, σ_{n} + τ_{n, [λ n]}^{<}) \\ \leq \bar{d} (u_{n}, τ_{n, [λ n]}^{<}) + \bar{d} (τ_{n, [λ n]}^{<}, σ_{n}) \\ = \bar{d} (u_{n}, τ_{n, [λ n]}^{<}) + \bar{d} (τ_{n, [λ n]}^{<} + σ_{[λ n]}, σ_{n} + σ_{[λ n]}) \\ \leq \bar{d} (u_{n}, τ_{n, [λ n]}^{<}) + \bar{d} (τ_{n, [λ n]}^{<}, σ_{[λ n]}) + \bar{d} (σ_{[λ n]}, σ_{n}) . \end{matrix}$ It follows by Lemma 2.2 (ii) that $\begin{matrix} \bar{d} (u_{n}, σ_{n}) & \leq & \bar{d} (u_{n}, τ_{n, [λ n]}^{<}) \\ + \frac{2 n - [λ n] + 1}{n - [λ n]} \bar{d} (σ_{n}, σ_{[λ n]}) . □ \end{matrix}$

3 Main results

We prove the following theorems.

Theorem 3.1.For a sequence (u_n) of fuzy numbers, let $\underset{λ \to 1^{+}}{lim inf} \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) = 0 .$ (12) If $\underset{n \to \infty}{lim sup} \bar{d} (σ_{n}, σ_{[λ n]}) \leq C (λ - 1)^{α} (λ > 1)$ (13) is satisfied for some constant C and some α > 1, then $lim_{n \to \infty} \bar{d} (u_{n}, σ_{n}) = 0 .$

Theorem 3.2.For a sequence (u_n) of fuzzy numbers, let $\underset{λ \to 1^{-}}{lim inf} \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, τ_{n, [λ n]}^{<}) = 0 .$ (14) If $\underset{n \to \infty}{lim sup} \bar{d} (σ_{n}, σ_{[λ n]}) \leq K (1 - λ)^{β} (0 < λ < 1) .$ (15) holds for some constant K and some β > 1, then $lim_{n \to \infty} \bar{d} (u_{n}, σ_{n}) = 0 .$

In Theorems 3.1 and 3.2, $τ_{n, [λ n]}^{>}$ and $τ_{n, [λ n]}^{<}$ are de la Vallée Poussin means of (u_n) and are defined by $τ_{n, [λ n]}^{>} : = \frac{1}{[λ n] - n} \sum_{k = n + 1}^{[λ n]} u_{k}$ for λ > 1 and sufficiently large n, and $τ_{n, [λ n]}^{<} : = \frac{1}{n - [λ n]} \sum_{k = [λ n] + 1}^{n} u_{k}$ for 0 < λ < 1 and sufficiently large n, respectively.

4 Proof of Theorem 3.1

Suppose that the condition (13) is satisfied. Taking the lim sup of both sides of (10) as n→ ∞, we obtain, for each λ > 1, $\begin{matrix} \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, σ_{n}) & \leq & \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) \\ + \frac{3 λ - 2}{λ - 1} \underset{n \to \infty}{lim sup} \bar{d} (σ_{n}, σ_{[λ n]}) \end{matrix}$

By condition (13), we have

$\begin{matrix} \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, σ_{n}) & \leq & \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) \\ + C (3 λ - 2) (λ - 1)^{α - 1} \end{matrix}$ (16) Taking the lim inf of both sides of (16) as λ → 1⁺, we get $\underset{n \to \infty}{lim sup} \bar{d} (u_{n}, σ_{n}) \leq \underset{λ \to 1^{+}}{lim inf} \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) .$ (17)

By (12), we have, from (17), $\underset{n \to \infty}{lim sup} \bar{d} (u_{n}, σ_{n}) \leq 0 .$ This completes the proof of Theorem 3.1.

5 Proof of Theorem 3.2

Suppose that the condition (13) is satisfied. Taking the lim sup of both sides of (10) as n→ ∞, we obtain, for each 0 < λ < 1,

$\begin{matrix} \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, σ_{n}) \leq \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, τ_{n, [λ n]}^{<}) \\ + \underset{n \to \infty}{lim sup} \frac{2 n - [λ n] + 1}{n - [λ n]} \bar{d} (τ_{n, [λ n]}^{<}, σ_{[λ n]}) . \end{matrix}$ (18)

By the condition (15), we have

$\begin{matrix} \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, σ_{n}) \leq \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, τ_{n, [λ n]}^{<}) \\ + K (2 - λ) (1 - λ)^{β - 1} \underset{n \to \infty}{lim sup} \bar{d} (τ_{n, [λ n]}^{<}, σ_{[λ n]}) . \end{matrix}$ (19) Taking the lim inf of both sides of (19) as λ → 1^-, we get

$\begin{matrix} \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, σ_{n}) \\ \leq \underset{λ \to 1^{+}}{lim inf} \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) . \end{matrix}$ (20) By (14), we have, from (20), $\underset{n \to \infty}{lim sup} \bar{d} (u_{n}, σ_{n}) \leq 0 .$ (21) This completes the proof of Theorem 3.2.

6 Corollaries

As corollaries we have the following classical Tauberian theorems for Cesàro summability of fuzzy numbers.

Corollary 6.1. Let (u_n) be Cesàro summable to L. If the condition (12) is satisfied, then (u_n) is convergent to L.

Proof. We show that the condition (13) is satisfied. Assume that (u_n) is Cesàro summable to L and the condition (12) is satisfied. We have, for λ > 1,

$\begin{matrix} \bar{d} (σ_{n}, σ_{[λ n]}) & = & \bar{d} (σ_{n} + L, σ_{[λ n]} + L) \\ \leq & \bar{d} (σ_{n}, L) + \bar{d} (σ_{[λ n]}, L) . \end{matrix}$ (22) It follows from (22) that $\underset{n \to \infty}{lim sup} \bar{d} (σ_{n}, σ_{[λ n]}) = 0 .$ This shows that the condition (13) is satisfied.

By Theorem 3.1, we have $lim_{n \to \infty} \bar{d} (u_{n}, σ_{n}) = 0 .$ Since (u_n) is Cesàro summable to L, we have, by the triangle inequality, $\begin{matrix} \bar{d} (u_{n}, L) & = & \bar{d} (u_{n} + σ_{n}, L + σ_{n}) \\ \leq & \bar{d} (u_{n}, σ_{n}) + \bar{d} (σ_{n}, L) . \end{matrix}$ This implies that (u_n) is convergent to L. □

Corollary 6.2. Let (u_n) be Cesàro summable to L. If the condition (12) is satisfied, then (u_n) is convergent to L.

Proof. We show that the condition (15) is satisfied. Assume that (u_n) is Cesàro summable to L and the condition (14) is satisfied. We have, for 0 < λ < 1,

$\begin{matrix} \bar{d} (σ_{n}, σ_{[λ n]}) & = & \bar{d} (σ_{n} + L, σ_{[λ n]} + L) \\ \leq & \bar{d} (σ_{n}, L) + \bar{d} (σ_{[λ n]}, L) . \end{matrix}$ (23) It follows from (23) that $\underset{n \to \infty}{lim sup} \bar{d} (σ_{n}, σ_{[λ n]}) = 0 .$ This shows that the condition (15) is satisfied.

By Theorem 3.1, we have $lim_{n \to \infty} \bar{d} (u_{n}, σ_{n}) = 0 .$ Since (u_n) is Cesàro summable to L, we have $\begin{matrix} \bar{d} (u_{n}, L) & = & \bar{d} (u_{n} + σ_{n}, L + σ_{n}) \\ \leq & \bar{d} (u_{n}, σ_{n}) + \bar{d} (σ_{n}, L) . \end{matrix}$ This implies that (u_n) is convergent to L. □

In the next corollary, it is shown that slow oscillation of a sequence is a Tauberian condition for Cesàro summability of sequences of fuzzy numbers.

Corollary 6.3. Let (u_n) be Cesàro summable to L. If (u_n) is slowly oscillating, then (u_n) is convergent to L.

Proof. Suppose that (u_n) is slowly oscillating. Then we have

$\begin{matrix} \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) \\ = \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) \\ \leq \bar{d} (\frac{1}{[λ n] - n} \sum_{k = n + 1}^{[λ n]} u_{n}, \frac{1}{[λ n] - n} \sum_{k = n + 1}^{[λ n]} u_{k}) \\ \leq \frac{1}{[λ n] - n} \sum_{k = n + 1}^{[λ n]} \bar{d} (u_{n}, u_{k}) \\ \leq max_{n + 1 \leq k \leq [λ n]} \bar{d} (u_{k}, u_{n}) . \end{matrix}$ (24) Taking the lim sup of both sides of (24) as n→ ∞, we get

$\begin{matrix} \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) \\ \leq \underset{n \to \infty}{lim sup} max_{n + 1 \leq k \leq [λ n]} \bar{d} (u_{k}, u_{n}) . \end{matrix}$ (25) It follows from (23) that $lim_{λ \to 1^{+}} \underset{n \to \infty}{lim sup} \bar{d} (u_{n}, τ_{n, [λ n]}^{>}) = 0 .$ By Corollary 6.1, we have that (u_n) converges to L. □

As a special case of Corollary 6.3 we have the following result.

Corollary 6.4.Let (u_n) be Cesàro summable to L. If $n \bar{d} (u_{n}, u_{n - 1}) = O (1),$ (26) then (u_n) is convergent to L.

Proof. The proof easily follows from the implication (refc3) ⇒ (refC) . □

7 An example

We now present an example defined by Talo and Çakan [18] for different purposes as an application of Theorem 3.1. Consider the fuzzy sequence u = (u_n) defined by $u_{n} = \sum_{k = 0}^{n} v_{k}$ , where $v_{k} (t) = {\begin{matrix} 1 - (k + 1) t & , if 0 \leq t \leq \frac{1}{k + 1} \\ 0 & , otherwise . \end{matrix}$ We first show that u = (u_n) is slowly oscillating. Indeed, for all n₁ < k ≤ [λn] with 1 < λ ≤ 1 + ɛ, we have $\begin{matrix} \bar{d} (u_{k}, u_{n}) & \leq & \sum_{j = n + 1}^{k} \bar{d} (v_{j}, 0) \\ = & \sum_{j = n + 1}^{k} \frac{1}{j + 1} \leq \frac{k}{n} - 1 \leq λ - 1 \leq ɛ . \end{matrix}$ We conclude by Corollary 6.3 that (u_n) satisfies condition (12). It is easy to show that (u_n) is not convergent. It follows by Corollary 6.3 that (σ_n) is not convergent. We now show that (13) is satisfied. For λ > 1, we have

$\begin{matrix} \bar{d} (σ_{n}, σ_{[λ n]}) \\ = \bar{d} (\frac{1}{[λ n] + 1} \sum_{j = 0}^{n} v_{j}, \frac{1}{n + 1} \sum_{j = 0}^{n} v_{j}) \\ + \bar{d} (\frac{1}{[λ n] + 1} \sum_{j = n + 1}^{[λ n]} v_{j}, 0) \\ \leq \frac{[λ n] - n}{[λ n] n} \sum_{j = 0}^{n} \bar{d} (v_{j}, 0) + \frac{1}{[λ n] + 1} \sum_{j = n + 1}^{[λ n]} \bar{d} (v_{j}, 0) \\ \leq \frac{[λ n] - n}{[λ n] n} \sum_{j = 0}^{n} \frac{1}{j + 1} + \frac{1}{[λ n] + 1} \sum_{j = n + 1}^{[λ n]} \frac{1}{j + 1} \\ \leq C (\frac{[λ n] - n}{[λ n]} \frac{log n}{n} + \frac{1}{[λ n] + 1} \frac{log [λ n]}{n}) \end{matrix}$ (27) for some constant C. Taking the lim sup of both sides of (27) as n→ ∞, we get $\underset{n \to \infty}{lim sup} \bar{d} (σ_{n}, σ_{[λ n]}) = 0 .$ Since conditions (12) and (13) of Theorem 3.1 are satisfied, we have $lim_{n \to \infty} \bar{d} (u_{n}, σ_{n}) = 0 .$

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