Let $(\Omega, \mathcal{F}, \mathbb{P}$ be a probability space, and consider a set $\mathcal{J}$ which is (at most) countable. For every $j \in \mathcal{J}$ we are given two events $B_j \subseteq A_j$ in $\mathcal{F}$. Show that \[ \mathbb{P} \left( \bigcup_{j \in \mathcal{J}} A_j ight) - \mathbb{P} \left( \bigcup_{j \in \mathcal{J}} B_j ight) \leq \sum_{j \in \mathcal{J}} [\mathbb{P}(A_j) - \mathbb{P}(B_j)] \]