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The inequality (3.4) yields 

It can be verified by applying the discrete analogue of Gronwall inequality that 

Thus, for n$_{1}$ − E ≤ n ≤ n$_{2}$ , we have 

The last inequality implies that 

One can confirm using (3.3) that ( ‖ A ‖ + L$_{f}$ ) n − n 1 + E +1 ≤ γϵ for n$_{1}$ ≤ n ≤ n$_{2}$ . Therefore, for each k ≥ k$_{0}$ , the inequality 

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is valid for n$_{1}$ ≤ n ≤ n$_{2}$ . Hence, ‖ φ$_{n}$$_{+}$$_{ζ}$$_{k}$ − φ$_{n}$ ‖ → 0 uniformly as k → ∞ for each n with n$_{1}$ ≤ n ≤ n$_{2}$ . 

Next, we will show the existence of a positive number ϵ$_{0}$ and a sequence $^{{}$˜ η$_{k}$ } with ˜ η$_{k}$ → ∞ as k → ∞ such that ‖ φ$_{ζ}$$_{k}$$_{+}$$_{˜}$ η$_{k}$ − φ$_{˜}$ $_{η$_{k}$}$‖ ≥ ϵ$_{0}$ for each k ∈ N . 

Using the relations 

and 

we obtain for k ∈ N that 

Therefore, 

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