Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers. Define $$B_j = \left\{\max_{\phi(j)+1 \leqslant n \leqslant \phi(j+1)} \left|S_n - S_{\phi(j)}\right| > 2^{-j}\right\},$$ $$B_{j,m} = \left\{\left|S_m - S_{\phi(j)}\right| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \quad \left|S_n - S_{\phi(j)}\right| \leqslant 2^{-j}\right\}.$$ 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 $\alpha = \pm 1$.