= Reverse Fatou inequality for events
{title2=$\mathbb P(\limsup_n A_n)\geq\limsup_n\mathbb P(A_n)$}
For any <sequence> of <events>, its <limsup of events> satisfies
$$
\mathbb P(\limsup_n A_n)\geq\limsup_n\mathbb P(A_n).
$$
Set $C_N=\bigcup_{n\geq N}A_n$. These are decreasing <events> with <intersection> $\limsup_n A_n$. <Continuity of probability> from above gives
$$
\mathbb P(\limsup_n A_n)=\lim_N\mathbb P(C_N)
\geq\lim_N\sup_{n\geq N}\mathbb P(A_n).
$$
This 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>.
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