Theorem (Countable Sub-additivity of Events)

Let $A_n, n\in\mathbb{N}$ be a sequence of events that are not necessarily disjoint. For a fixed n: $$P(A_1\cap\ldots\cap A_n)\le\sum_{i=1}^n P(A_i)$$ Countable sub-additivity: $$P(\bigcup_{n=1}^\infty A_n)\le\sum_{n=1}^\infty P(A_n)$$