question with space of sequences

Question

Let $C_s^N$ to be the space of bi-directional sequences f which f[m+N] = f[m] . For each $f ∈ C_s^N$ , we define $∇_+ f [m] = f[m+1] – f[m]$ . for p>1 , we define $∇_+^p = ∇_+ (∇_+^{p-1})$

1. Calculate $∇_+^2f[m]$
2. Show that if $f ∈ C_s^N$ and exist p∈N which hold $∇_+^p f≡ 0$ , so $f≡ c$ for some constant c