3 Unpredictable sequences The following theorem is concerned with the existence of an unpredictable solution of the discrete equa- tion (2.1). Theorem 3.1 The bounded solution { φ$_{n}$ } , n ∈ Z , of equation (2.1) is unpredictable under the conditions ( C 1) − ( C 3) . Proof. Fix an arbitrary positive number ϵ, and suppose that γ is a positive number satisfying Let n$_{1}$ and n$_{2}$ be integers such that n$_{2}$ > n$_{1}$ , and take a natural number E with Since { ψ$_{n}$ } , n ∈ Z , is an unpredictable sequence, there exist a positive number ϵ$_{0}$ and sequences { ζ$_{k}$ } , { η$_{k}$ } , k ∈ N , of positive integers both of which diverge to infinity such that ‖ ψ$_{n}$$_{+}$$_{ζ}$$_{k}$ − ψ$_{n}$ ‖ → 0 uniformly as k → ∞ for each n with n$_{1}$ − E − 1 ≤ n ≤ n$_{2}$ − 1 and ‖ ψ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$ − ψ$_{η}$$_{k}$ ‖ ≥ ϵ$_{0}$ for each k ∈ N . First of all, we will show that ‖ φ$_{n}$$_{+}$$_{ζ}$$_{k}$ − φ$_{n}$ ‖ → 0 uniformly as k → ∞ for each n with n$_{1}$ ≤ n ≤ n$_{2}$ . There exists a natural number k$_{0}$, independent of n, such that for each k ≥ k$_{0}$ the inequality ‖ ψ$_{n}$$_{+}$$_{ζ}$$_{k}$ − ψ$_{n}$ ‖ < γϵ is valid whenever n$_{1}$ − E − 1 ≤ n ≤ n$_{2}$ − 1 . Fix an arbitrary integer k ≥ k$_{0}$ . One can obtain using the relation (2.2) that Therefore, for n$_{1}$ − E ≤ n ≤ n$_{2}$, we have Let us denote and φ$_{n}$$_{+}$$_{ζ}$$_{k}$ − φ$_{n}$ = n ∑ j = −∞ A n − j [ f ( φ$_{j}$$_{+}$$_{ζ}$$_{k}$ − $_{1}$) − f ( φ$_{j}$ − $_{1}$) + ψ$_{j}$$_{+}$$_{ζ}$$_{k}$ − 1 − ψ$_{j}$ − $_{1}$] . ‖ φ$_{n}$$_{+}$$_{ζ}$$_{k}$ − φ$_{n}$ ‖ < 2( M$_{f}$ + M$_{ψ}$ ) 1 − ‖ A ‖ ‖ A ‖ n − n 1 + E +1 + γϵ 1 − ‖ A ‖ ( 1 − ‖ A ‖ n − n 1 + E +1 ) + L$_{f}$ n ∑ j = n $_{1}$− E ‖ A ‖ n − j ‖ φ$_{j}$$_{+}$$_{ζ}$$_{k}$ − 1 − φ$_{j}$ − $_{1}$‖ . (3.4) z$_{n}$ = ‖ A ‖ − n ‖ φ$_{n}$$_{+}$$_{ζ}$$_{k}$ − φ$_{n}$ ‖ ϑ$_{n}$ = 2( M$_{f}$ + M$_{ψ}$ ) 1 − ‖ A ‖ ‖ A ‖ − n 1 + E +1 + γϵ 1 − ‖ A ‖ ( ‖ A ‖ − n − ‖ A ‖ − n 1 + E +1 ) . E ≥ ln( γϵ ) ln( ‖ A ‖ + L$_{f}$ ) − 1 . (3.3) γ ≤ [ 1 1 − ‖ A ‖ − L$_{f}$ + 2( M$_{f}$ + M$_{ψ}$ ) 1 − ‖ A ‖ ] − 1 . 4