With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), prove that if the event $B_j$ occurs, then there exist $m \in \llbracket \phi(j)+1, \phi(j+1) \rrbracket$ and $\alpha \in \{-1, +1\}$ such that the event $$\left\{\left|\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}\right| > 2^{-j}\right\} \cap B_{j,m}$$ also occurs. One may express $S_m - S_{\phi(j)}$ in terms of the two numbers $\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}$ with $\alpha = \pm 1$.
With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), prove that if the event $B_j$ occurs, then there exist $m \in \llbracket \phi(j)+1, \phi(j+1) \rrbracket$ and $\alpha \in \{-1, +1\}$ such that the event
$$\left\{\left|\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}\right| > 2^{-j}\right\} \cap B_{j,m}$$
also occurs. One may express $S_m - S_{\phi(j)}$ in terms of the two numbers $\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}$ with $\alpha = \pm 1$.