Reverse Fatou inequality for events ($\mathbb{P}({\mathrm{limsup}}_n{A}_n)\ge {\mathrm{limsup}}_n\mathbb{P}(A_n)$)

For any sequence of events, its limsup of events satisfiesSet $C_N={\bigcup }_{n\ge N}{A}_n$. These are decreasing events with intersection ${\mathrm{limsup}}_n{A}_n$. Continuity of probability from above givesThis is also the bounded-indicator form of the reversed Fatou lemma. The inequality can be strict: independent fair-coin head events each have probability $1/2$, while their limsup of events has probability $1$ by the Second Borel-Cantelli lemma.