Infinitely often and almost always

Let $A_1,A_2,A_3,\dots$ be a sequence of events in some sample space $\mathrm{\Omega }$.

perspective	infinitely often	almost always
unions and intersections	$x\in \bigcap _{N=1}^{\mathrm{\infty }}\bigcup _{n=N}^{\mathrm{\infty }}{A}_n$	$x\in \bigcup _{N=1}^{\mathrm{\infty }}\bigcap _{n=N}^{\mathrm{\infty }}{A}_n$
first-order quantifiers	$\mathrm{\forall }N\ge 1\mathrm{\exists }n\ge Nx\in A_n$	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	$x\in {lim sup}_{n\to \mathrm{\infty }}{A}_n$	$x\in {lim inf}_{n\to \mathrm{\infty }}{A}_n$
limit of sup/inf	$x\in \underset{N\to \mathrm{\infty }}{lim}{sup}_{n=N}{A}_n$	$x\in \underset{N\to \mathrm{\infty }}{lim}{inf}_{n=N}{A}_n$