We have $d_{n+1}(x) = 2^{n+1}(\pi_{n+1}(x) - \pi_n(x))$ for all $(x,n) \in \mathbb{R} \times \mathbb{N}$. Establish $$\forall (x,j) \in \mathbb{R} \times \mathbb{N}^{\star}, \quad d_j(x) \in \{0,1\}.$$