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 the events $$A_j = \left\{\left|S_{\phi(j+1)} - S_{\phi(j)}\right| > 2^{-j}\right\},$$ $$B_{j,m} = \left\{\left|S_m - S_{\phi(j)}\right| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \quad \left|S_n - S_{\phi(j)}\right| \leqslant 2^{-j}\right\}.$$ Explain how to deduce the formula $\mathbb{P}(A_j) = \sum_{m=\phi(j)+1}^{\phi(j+1)} \mathbb{P}(A_j \cap B_{j,m})$.
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 the events
$$A_j = \left\{\left|S_{\phi(j+1)} - S_{\phi(j)}\right| > 2^{-j}\right\},$$
$$B_{j,m} = \left\{\left|S_m - S_{\phi(j)}\right| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \quad \left|S_n - S_{\phi(j)}\right| \leqslant 2^{-j}\right\}.$$
Explain how to deduce the formula $\mathbb{P}(A_j) = \sum_{m=\phi(j)+1}^{\phi(j+1)} \mathbb{P}(A_j \cap B_{j,m})$.