With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), deduce that $$\mathbb{P}(B_j) \leqslant 2\mathbb{P}(A_j).$$
With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), deduce that
$$\mathbb{P}(B_j) \leqslant 2\mathbb{P}(A_j).$$