Infinitely often and almost always

Let $A_{1},A_{2},A_{3},\ldots$ be a sequence of events in some sample space $\Omega$.

perspective	infinitely often	almost always
unions and intersections	$x\in \bigcap _{N=1}^{\infty }\bigcup _{n=N}^{\infty }A_{n}$	$x\in \bigcup _{N=1}^{\infty }\bigcap _{n=N}^{\infty }A_{n}$
first-order quantifiers	$\forall N\geq 1\ \exists n\geq N\ x\in A_{n}$	Failed to parse (unknown function "\eq"): {\displaystyle \exists N \eq 1\ \forall n \geq N\ x \in A_n}
verbal expression
lim sup/lim inf	$x\in \limsup _{n\to \infty }A_{n}$	$x\in \liminf _{n\to \infty }A_{n}$
limit of sup/inf	$x\in \lim _{N\to \infty }\sup _{n=N}A_{n}$	$x\in \lim _{N\to \infty }\inf _{n=N}A_{n}$