On some new sequence spaces defined by infinite matrix and modulus

Abstract

The goal of this paper is to introduce and study some properties of some sequence spaces that are defined using the φ-function and the generalized three parametric real matrix A. Also, we define A-statistical convergence.

MSC:40H05, 40C05.

1 Introduction and background

Let s denote the set of all real and complex sequences $x=(x_k)$. By $l_{\mathrm{\infty }}$ and c, we denote the Banach spaces of bounded and convergent sequences $x=(x_k)$ normed by $\parallel x\parallel ={sup}_n|x_n|$, respectively. A linear functional L on $l_{\mathrm{\infty }}$ is said to be a Banach limit [1] if it has the following properties:

• (1) $L(x)\ge 0$ if $n\ge 0$ (i.e., $x_n\ge 0$ for all n),

• (2) $L(e)=1$, where $e=(1,1,\dots )$,

• (3) $L(Dx)=L(x)$, where the shift operator D is defined by $D(x_n)=\{x_{n+1}\}$.

Let B be the set of all Banach limits on $l_{\mathrm{\infty }}$. A sequence $x\in {\ell }_{\mathrm{\infty }}$ is said to be almost convergent if all Banach limits of x coincide. Let $\hat{c}$ denote the space of almost convergent sequences. Lorentz [2] has shown that

$$\hat{c}=\{x\in l_{\mathrm{\infty }}:\underset{m}{lim}{t}_{m,n}(x)\text{ exists uniformly in }n\},$$

where

$$t_{m,n}(x)=\frac{x_n+x_{n+1}+x_{n+2}+\cdots +x_{n+m}}{m+1}.$$

By a lacunary $\theta =(k_r)$, $r=0,1,2,\dots$ , where $k_0=0$, we shall mean an increasing sequence of non-negative integers with $k_r-k_{r-1}\to \mathrm{\infty }$ as $r\to \mathrm{\infty }$. The intervals determined by θ will be denoted by $I_r=(k_{r-1},k_r]$ and $h_r=k_r-k_{r-1}$.The ratio $\frac{k_r}{k_{r-1}}$ will be denoted by $q_r$.

The space of lacunary strongly convergent sequences $N_{\theta }$ was defined by Freedman et al. [3] as follows:

$$N_{\theta }=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{k\in I_r}|x_k-le|=0\text{ for some }l\}.$$

There is a strong connection between $N_{\theta }$ and the space w of strongly Cesàro summable sequences which is defined by

$$w=\{x=(x_k):\underset{n}{lim}\frac{1}{n}\sum _{k=0}^n|x_k-le|=0\text{ for some }l\}.$$

In the special case where $\theta =(2^r)$, we have $N_{\theta }=w$.

More results on lacunary strong convergence can be seen from [4–11].

Ruckle [12] used the idea of a modulus function f to construct a class of FK spaces

$$L(f)=\{x=(x_k):\sum _{k=1}^{\mathrm{\infty }}f(|x_k|)<\mathrm{\infty }\}.$$

The space $L(f)$ is closely related to the space $l_1$ which is an $L(f)$ space with $f(x)=x$ for all real $x\ge 0$.

Maddox [13] introduced and examined some properties of the sequence spaces $w_0(f)$, $w(f)$ and $w_{\mathrm{\infty }}(f)$ defined using a modulus f, which generalized the well-known spaces $w_0$, w and $w_{\mathrm{\infty }}$ of strongly summable sequences.

Recently Savaş [14] generalized the concept of strong almost convergence by using a modulus f and examined some properties of the corresponding new sequence spaces. Waszak [15] defined the lacunary strong $(A,\phi )$-convergence with respect to a modulus function.

Following Ruckle, a modulus function f is a function from $[0,\mathrm{\infty })$ to $[0,\mathrm{\infty })$ such that

• (i) $f(x)=0$ if and only if $x=0$,

• (ii) $f(x+y)\le f(x)+f(x)$ for all $x,y\ge 0$,

• (iii) f increasing,

• (iv) f is continuous from the right at zero.

Since $|f(x)-f(y)|\le f(|x-y|)$, it follows from condition (iv) that f is continuous on $[0,\mathrm{\infty })$.

By a φ-function we understood a continuous non-decreasing function $\phi (u)$ defined for $u\ge 0$ and such that $\phi (0)=0$, $\phi (u)>0$ for $u>0$ and $\phi (u)\to \mathrm{\infty }$ as $u\to \mathrm{\infty }$.

A φ-function φ is called no weaker than a φ-function ψ if there are constants $c,b,k,l>0$ such that $c\psi (lu)\le b\phi (ku)$ (for all large u) and we write $\psi \prec \phi$.

φ-functions φ and ψ are called equivalent and we write $\phi \sim \psi$ if there are positive constants $b_1$, $b_2$, c, $k_1$, $k_2$, l such that $b_1\phi (k_{1}u)\le c\psi (lu)\le b_2\phi (k_{2}u)$ (for all large u).

A φ-function φ is said to satisfy $({\mathrm{\Delta }}_2)$-condition (for all large u) if there exists a constant $K>1$ such that $\phi (2u)\le K\phi (u)$.

In the present paper, we introduce and study some properties of the following sequence space that is defined using the φ-function and the generalized three parametric real matrix.

2 Main results

Let φ and f be a given φ-function and a modulus function, respectively. Moreover, let $\mathbf{A}=(a_{nk}(i))$ be the generalized three parametric real matrix, and let a lacunary sequence θ be given. Then we define

$$N_{\theta }^0(\mathbf{A},\phi ,f)=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{n\in I_r}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)=0\text{ uniformly in }i\}.$$

If $x\in N_{\theta }^0(\mathbf{A},\phi ,f)$, the sequence x is said to be lacunary strong $(\mathbf{A},\phi )$-convergent to zero with respect to a modulus f. When $\phi (x)=x$, for all x, we obtain

$$N_{\theta }^0(\mathbf{A},f)=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{n\in I_r}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)(|x_k|)\left|\right)=0\text{ uniformly in }i\}.$$

If we take $f(x)=x$, we write

$$N_{\theta }^0(\mathbf{A},\phi )=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{n\in I_r}|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)|=0\text{ uniformly in }i\}.$$

If we take $\mathbf{A}=I$ and $\phi (x)=x$ respectively, then we have [16]

$$N_{\theta }^0=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{n\in I_r}f(|x_k|)=0\text{ uniformly in }i\}.$$

If we define the matrix $A=(a_{nk}(i))$ as follows: for all i,

$$a_{nk}(i):=\{\begin{array}{ll}\frac{1}{n}& \text{if }n\ge k,\\ 0& \text{otherwise},\end{array}$$

then we have

$$\begin{array}{c}{N}_{\theta }^0(\mathbf{C},\phi ,f)=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{n\in I_r}f\left(\right|\frac{1}{n}\sum _{k=1}^n\phi (|x_k|)\left|\right)=0\text{ uniformly in }i\},\hfill \\ a_{nk}(i):=\{\begin{array}{ll}\frac{1}{n}& \text{if }i\le k\le i+n-1,\\ 0& \text{otherwise},\end{array}\hfill \end{array}$$

then we have

$$N_{\theta }^0(\hat{c},\phi ,f)=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{n\in I_r}f\left(\right|\frac{1}{n}\sum _{k=i}^{i+n}\phi (|x_k|)\left|\right)=0\text{ uniformly in }i\}.$$

We are now ready to write the following theorem.

Theorem 2.1 Let $\mathbf{A}=(a_{nk}(i))$ be the generalized three parametric real matrix, and let the φ-function $\phi (u)$ satisfy the condition $({\mathrm{\Delta }}_2)$. Then the following conditions are true.

• (a) If $x=(x_k)\in w(\mathbf{A},\phi ,f)$ and α is an arbitrary number, then $\alpha x\in w(\mathbf{A},\phi ,f)$.

• (b) If $x,y\in w(\mathbf{A},\phi ,f)$, where $x=(x_k)$, $y=(y_k)$ and α, β are given numbers, then $\alpha x+\beta y\in w(\mathbf{A},\phi ,f)$.

The proof is a routine verification by using standard techniques and hence is omitted.

Theorem 2.2 Let f be any modulus function, and let the generalized three parametric real matrix A and the sequence θ be given. If

$$w(\mathbf{A},\phi ,f)=\{x=(x_k):\underset{m}{lim}\frac{1}{m}\sum _{n=1}^{m}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)=0\mathit{\text{ uniformly in }}i\},$$

then the following relations are true.

• (a) If ${lim\hspace{0.17em}inf}_r{q}_r>1$, then we have $w(A,\phi ,f)\subseteq N_{\theta }^0(\mathbf{A},\phi ,f)$.

• (b) If ${sup}_r{q}_r<\mathrm{\infty }$, then we have $N_{\theta }^0(\mathbf{A},\phi ,f)\subseteq w(A,\phi ,f)$.

• (c) $1<{lim\hspace{0.17em}inf}_r{q}_r\le {lim\hspace{0.17em}sup}_r{q}_r<\mathrm{\infty }$, then we have $N_{\theta }^0(\mathbf{A},\phi ,f)=w(\mathbf{A},\phi ,f)$.

Proof (a) Let us suppose that $x\in w(A,\phi ,f)$. There exists $\delta >0$ such that $q_r>1+\delta$ for all $r\ge 1$, and we have $h_r/k_r\ge \delta /(1+\delta )$ for sufficiently large r. Then, for all i,

$$\begin{array}{r}\frac{1}{k_r}\sum _{n=1}^{k_r}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)\\ \ge \frac{1}{k_r}\sum _{n\in I_r}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)\\ =\frac{h_r}{k_r}\frac{1}{h_r}\sum _{n\in I_r}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)\\ \ge \frac{\delta }{1+\delta }\frac{1}{h_r}\sum _{n\in I_r}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\phi (|x_k|)\left|\right).\end{array}$$

Hence, $x\in N_{\theta }^0(\mathbf{A},\phi ,f)$.

• (b) If ${lim\hspace{0.17em}sup}_r{q}_r<\mathrm{\infty }$, then there exists $M>0$ such that $q_r<M$ for all $r\ge 1$. Let $x\in N_{\theta }^0(\mathbf{A},\phi ,f)$ and ε be an arbitrary positive number, then there exists an index $j_0$ such that for every $j\ge j_0$ and all i,

  $$R_j=\frac{1}{h_j}\sum _{n\in I_r}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)<\epsilon .$$

Thus, we can also find $K>0$ such that $R_j\le K$ for all $j=1,2,\dots$ . Now, let m be any integer with $k_{r-1}\le m\le k_r$, then we obtain, for all i,

$$I=\frac{1}{m}\sum _{n=1}^{m}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)\le \frac{1}{k_{r-1}}\sum _{n=1}^{k_r}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)=I_1+I_2,$$

where

$$\begin{array}{c}{I}_1=\frac{1}{k_{r-1}}\sum _{j=1}^{j_0}\sum _{n\in I_j}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right),\hfill \\ I_2=\frac{1}{k_{r-1}}\sum _{j=j_{0+1}}^m\sum _{n\in I_j}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right).\hfill \end{array}$$

It is easy to see that

$$\begin{array}{rcl}{I}_1& =& \frac{1}{k_{r-1}}\sum _{j=1}^{j_0}\sum _{n\in I_j}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)\\ =& \frac{1}{k_{r-1}}(\sum _{n\in I_1}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)+\cdots +\sum _{n\in I_{j_0}}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right))\\ \le & \frac{1}{k_{r-1}}(h_1{R}_1+\cdots +h_{j_0}{R}_{j_0})\\ \le & \frac{1}{k_{r-1}}{j}_0{k}_{j_0}\underset{1\le i\le j_0}{sup}{R}_i\\ \le & \frac{j_0{k}_{j_0}}{k_{r-1}}K.\end{array}$$

Moreover, we have, for all i,

$$\begin{array}{rcl}{I}_2& =& \frac{1}{k_{r-1}}\sum _{j=j_0+1}^m\sum _{n\in I_j}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\phi (|x_k|)\left|\right)\\ =& \frac{1}{k_{r-1}}\sum _{j=j_0+1}^m\frac{1}{h_j}\sum _{n\in I_j}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\phi (|x_k|)\left|\right)h_j\\ \le & \epsilon \frac{1}{k_{r-1}}\sum _{j=j_0+1}^m{h}_j\\ \le & \epsilon \frac{k_r}{k_{r-1}}\\ =& \epsilon q_r<\epsilon \cdot M.\end{array}$$

Thus $I\le \frac{j_0{k}_{j_0}}{k_{r-1}}K+\epsilon \cdot M$. Finally, $x\in w(A,\psi ,f)$.

The proof of (c) follows from (a) and (b). This completes the proof. □

We now prove the following theorem.

Theorem 2.3 Let f be a modulus function. Then $N_{\theta }^0(A,\phi )\subset N_{\theta }^0(A,\phi ,f)$.

Proof Let $x\in N_{\theta }^0(A,\phi )$. Let $\epsilon >0$ be given and choose $0<\delta <1$ such that $f(x)<\epsilon$ for every $x\in [0,\delta ]$. We can write

$$\frac{1}{h_r}\sum _{n\in I_r}f|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)|=S_1+S_2,$$

where $S_1=\frac{1}{h_r}{\sum }_{n\in I_r}f(|{\sum }_{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)|)$, and this sum is taken over

$$|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|\left)\right|\le \delta$$

and

$$S_2=\frac{1}{h_r}\sum _{n\in I_r}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right),$$

and this sum is taken over

$$|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)|>\delta .$$

By the definition of the modulus f, we have $S_1=\frac{1}{h_r}{\sum }_{n\in I_r}f(\delta )=f(\delta )<\epsilon$ and further

$$S_2=f(1)\frac{1}{\delta }\frac{1}{h_r}\sum _{n\in I_r}\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|).$$

Therefore we have $x\in N_{\theta }^0(\mathbf{A},\phi ,f)$.

This completes the proof. □

3 A-Statistical convergence

The idea of convergence of a real sequence was extended to statistical convergence by Fast [17] (see also Schoenberg [18]) as follows: If ℕ denotes the set of natural numbers and $K\subset \mathbb{N}$, then $K(m,n)$ denotes the cardinality of the set $K\cap [m,n]$. The upper and lower natural density of the subset K is defined by

$$\overline{d}(K)=\underset{n\to \mathrm{\infty }}{lim}sup\frac{K(1,n)}{n}\text{and}\underline{d}(K)=\underset{n\to \mathrm{\infty }}{lim}inf\frac{K(1,n)}{n}.$$

If $\overline{d}(K)=\underline{d}(K)$, then we say that the natural density of K exists and it is denoted simply by $d(K)$. Clearly, $d(K)={lim}_{n\to \mathrm{\infty }}\frac{K(1,n)}{n}$.

A sequence $(x_k)$ of real numbers is said to be statistically convergent to L if for arbitrary $\epsilon >0$, the set $K(\epsilon )=\{k\in \mathbb{N}:|x_k-L|\ge \epsilon \}$ has natural density zero. Statistical convergence turned out to be one of the most active areas of research in summability theory after the work of Fridy [19] and Šalát [20].

In another direction, a new type of convergence, called lacunary statistical convergence, was introduced in [21] as follows.

A sequence ${(x_k)}_{n\in \mathbb{N}}$ of real numbers is said to be lacunary statistically convergent to L (or $S_{\theta }$-convergent to L) if for any $\epsilon >0$,

$$\underset{r\to \mathrm{\infty }}{lim}\frac{1}{h_r}\left|\{k\in I_r:|x_k-L|\ge \epsilon \}\right|=0,$$

where $|A|$ denotes the cardinality of $A\subset \mathbb{N}$. In [21] the relation between lacunary statistical convergence and statistical convergence was established among other things. Moreover, Kolk [22] defined A-statistical convergence by using non-negative regular summability matrix.

In this section we define $(A,\phi )$-statistical convergence by using the generalized three parametric real matrix and the φ-function $\phi (u)$.

Let θ be a lacunary sequence, and let $\mathbf{A}=(a_{nk}(i))$ be the generalized three parametric real matrix; let the sequence $x=(x_k)$, the φ-function $\phi (u)$ and a positive number $\epsilon >0$ be given. We write, for all i,

$$K_{\theta }^r((A,\phi ),\epsilon )=\{n\in I_r:\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\ge \epsilon \}.$$

The sequence x is said to be $(\mathbf{A},\phi )$-statistically convergent to a number zero if for every $\epsilon >0$,

$$\underset{r}{lim}\frac{1}{k_r}\mu \left(K_{\theta }^r((A,\phi ),\epsilon )\right)=0\text{uniformly in }n,$$

where $\mu (K_{\theta }^r((A,\phi ),\epsilon ))$ denotes the number of elements belonging to $K_{\theta }^r((\mathbf{A},\phi ),\epsilon )$. We denote by $S_{\theta }^0(\mathbf{A},\phi )$ the set of sequences $x=(x_k)$ which are lacunary $(\mathbf{A},\phi )$-statistical convergent to zero. We write

$$S_{\theta }^0(\mathbf{A},\phi )=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\mu \left(K_{\theta }^r((A,\phi ),\epsilon )\right)=0\text{ uniformly in }i\}.$$

Theorem 3.1 If $\psi \prec \phi$, then $S_{\theta }^0(A,\psi )\subset S_{\theta }^0(A,\phi )$.

Proof By assumption we have $\psi (|x_k|)\le b\phi (c|x_k|)$ and we have, for all i,

$$\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\psi (|x_k|)\le b\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (c|x_k|)\le L\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)$$

for $b,c>0$, where the constant L is connected with the properties of φ. Thus, the condition ${\sum }_{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\ge 0$ implies the condition ${\sum }_{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\ge \epsilon$, and finally we get

$$\mu \left(K_{\theta }^r((A,\phi ),\epsilon )\right)\subset \mu \left(K_{\theta }^r((A,\psi ),\epsilon )\right)$$

and

$$\underset{r}{lim}\frac{1}{h_r}\mu \left(K_{\theta }^r((A,\phi ),\epsilon )\right)\le \underset{r}{lim}\frac{1}{h_r}\mu \left(K_{\theta }^r((A,\psi ),\epsilon )\right).$$

This completes the proof. □

We finally prove the following theorem.

Theorem 3.2 (a) If the matrix A, the sequence θ and functions f and φ are given, then

$$N_{\theta }^0((A,\phi ),f)\subset S_{\theta }^0(A,\phi ).$$

(b) If the φ-function $\phi (u)$ and the matrix A are given, and if the modulus function f is bounded, then

$$S_{\theta }^0(A,\phi )\subset N_{\theta }^0((A,\phi ),f).$$

(c) If the φ-function $\phi (u)$ and the matrix A are given, and if the modulus function f is bounded, then

$$S_{\theta }^0(A,\phi )=N_{\theta }^0((A,\phi ),f).$$

Proof (a) Let f be a modulus function, and let ε be a positive number. We write the following inequalities:

$$\begin{array}{r}\frac{1}{h_r}\sum _{n\in I_r}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)\\ \ge \frac{1}{h_r}\sum _{n\in I_r^1}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)\\ \ge \frac{1}{h_r}f(\epsilon )\sum _{n\in I_r^1}1\\ \ge \frac{1}{h_r}f(\epsilon )\mu (K_{\theta }^r(A,\phi ),\epsilon ),\end{array}$$

where

$$I_r^1=\{n\in I_r:\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\ge \epsilon \}.$$

Finally, if $x\in N_{\theta }^0((A,\phi ),f)$, then $x\in S_{\theta }^0(A,\phi )$.

• (b) Let us suppose that $x\in S_{\theta }^0(A,\phi )$. If the modulus function f is a bounded function, then there exists an integer L such that $f(x)<L$ for $x\ge 0$. Let us take

  $$I_r^2=\{n\in I_r:\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)<\epsilon \}.$$

Thus we have

$$\begin{array}{r}\frac{1}{h_r}\sum _{n\in I_r}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)\\ \le \frac{1}{h_r}\sum _{n\in I_r^1}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)\\ +\frac{1}{h_r}\sum _{n\in I_r^2}f\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)\\ \le \frac{1}{h_r}M\mu \left(K_{\theta }^r((A,\phi ),\epsilon )\right)+f(\epsilon ).\end{array}$$

Taking the limit as $\epsilon \to 0$, we obtain that $x\in N_{\theta }^0(A,\phi ,f)$.

The proof of (c) follows from (a) and (b).

This completes the proof. □