A quick question on the limsup of sets and logical connectives

Question

I know we can interpret $x\in\limsup{A_n}$ iff $x$ is in infinitely many of the $A_n$'s, but am confused as to which logical connective we need to use to describe when $x\in\limsup{A_n}$, when we write it in a sentence of symbolic logic. So is it right to say that; $$x\in\limsup{A_n}\space\space\space\text{iff}\space\space\space(\forall n\in\mathbb{N})(\exists m\in\mathbb{N})(m>n\space \&\space x\in A_m)$$ or $$x\in\limsup{A_n}\space\space\space\text{iff}\space\space\space(\forall n\in\mathbb{N})(\exists m\in\mathbb{N})(m>n\Rightarrow x\in A_m)$$ or $$x\in\limsup{A_n}\space\space\space\text{iff}\space\space\space(\forall n\in\mathbb{N})(\exists m>n)(x\in A_m)\space \space ?$$ So would appreciate any feedback to let me know which statement (if any) is correct and if they are all wrong please can you give me the correct statement. Thanks in advance