Lemma 3.1 For each increasing sequence { m$_{n}$ } of positive integers, there exist a sequence s ∗∗ ∈ Σ$_{2}$ and sequences { α$_{n}$ } , { β$_{n}$ } of positive integers, both of which diverge to infinity, such that Proof. Fix an arbitrary increasing sequence { m$_{n}$ } of positive integers. For each n ∈ $^{N}$, define α$_{n}$ = ∑ n + m$_{n}$ k =1 k 2 k and β$_{n}$ = n + m$_{n}$ + 1 . Clearly, both of the sequences { α$_{n}$ } and { β$_{n}$ } diverge to infinity. We will construct a sequence s ∗∗ ∈ Σ$_{2}$ such that the inequalities ( i ) and ( ii ) are valid. First of all, we choose the terms s ∗∗ 0 , s ∗∗ 1 , . . . , s ∗∗ α$_{1}$ − 1 by successively placing the blocks of 0 ’s and 1 ’s in an increasing length, starting from the blocks of length 1 till the end of the ones with length m$_{1}$ + 1 . The order of the blocks with the same length can be arbitrary without any repetitions. Let us take s ∗∗ α$_{1}$ + k = s ∗∗ k for k = 0 , 1 , . . . , m$_{1}$ + 1 , i.e., the terms of the first block ( s ∗∗ $_{α$_{1}$}$s ∗∗ α$_{1}$ $_{+1}$· · · s ∗∗ α$_{1}$ + m$_{1}$ $_{+1}$) of length m$_{1}$ + 2 is chosen the same as the first m$_{1}$ + 2 terms of the sequence s $^{∗∗}$. Moreover, we take the second block of length m$_{1}$ + 2 in a such a way that its first term s ∗∗ α$_{1}$ + β$_{1}$ is different from s ∗∗ $_{β$_{1}$}$. After that we continue placing the remaining blocks of length m$_{1}$ + 2 and the ones with length greater than m$_{1}$ + 2 till the last block of length m$_{2}$ + 2 , and again, the blocks of the same length can be in any order without repetitions. For each n ≥ 2 , we set the last block of length m$_{n}$ + n such that s ∗∗ α$_{n}$ − k = s ∗∗ α$_{k}$ − k for each k = 1 , 2 , . . . , n − 1 . Then, the terms of the first block of length m$_{n}$ + n +1 are constituted by taking s ∗∗ α$_{n}$ + k = s ∗∗ k , k = 0 , 1 , . . . , m$_{n}$ + n. Moreover, the second block of length m$_{n}$ + n + 1 is chosen such that s ∗∗ α$_{n}$ + β$_{n}$ = s ∗∗ $_{β$_{n}$}$. Lastly, the remaining blocks of 0 ’s and 1 ’s are successively placed similar to the case mentioned above so that the lengths of the blocks in s ∗∗ are in an increasing order and there are no repetitions within the blocks of the same length. By this way the construction of the sequence s ∗∗ ∈ Σ$_{2}$ is completed. We fix the extension of the sequence s ∗∗ to the left by choosing σ − $^{k}$( s $^{∗∗}$)$_{0}$ = s ∗∗ α$_{k}$ − $_{k}$, k ∈ $^{N}$. For each n ∈ $^{N}$, we have σ α$_{n}$ − $^{n}$( s $^{∗∗}$)$_{k}$ = σ − $^{n}$( s $^{∗∗}$)$_{k}$ , k = 0 , 1 , . . . , m$_{n}$ + 2 n and σ α$_{n}$ + $^{β$_{n}$}$( s $^{∗∗}$)$_{0}$ = σ $^{β$_{n}$}$( s $^{∗∗}$)$_{0}$ so that the inequalities ( i ) and ( ii ) are valid. □ The technique presented in [5] can be used to show that the trajectory T ( s $^{∗∗}$) = { σ $^{i}$( s $^{∗∗}$) : i ∈ Z } is dense in Σ$_{2}$ , i.e., Θ$_{s}$ ∗∗ = Σ$_{2}$ . By Lemma 2.2 in [2] any sequence σ $^{i}$( s $^{∗∗}$) , i ∈ $^{Z}$, is an unpredictable point of the Bernoulli dynamics on Σ$_{2}$ , and Σ$_{2}$ is a quasi-minimal set. It implies from the last theorem that T ( s $^{∗∗}$) is an unpredictable function on $^{Z}$, i.e., an unpredictable sequence. According to Theorem 3.1 presented in [2], the dynamics on Σ$_{2}$ is Poincaré chaotic. Moreover, there are infinitely many unpredictable sequences in the set. 6 (ii) d ( σ α$_{n}$ + $^{β$_{n}$}$( s $^{∗∗}$) , σ $^{β$_{n}$}$( s $^{∗∗}$)) ≥ 1 for each n ∈ $^{N}$. (i) d ( σ α$_{n}$ + $^{r}$( s $^{∗∗}$) , σ $^{r}$( s $^{∗∗}$)) ≤ 2 − $^{m$_{n}$}$, r = − n, − n + 1 , . . . , n,