The liminf and limsup for sets

Liminf and limsup for sets will be used in the proof of Borel-Cantelli Lemma. There are two equivalent definitions on Wikipedia, and it seems weird at first. Namely: liminf (resp. limsup) is defined as union of intersections (intersection of unions), and is also said as occurring infinitely often (occurring for all but finite indices). Let us verify their equivalence.

Let a sequence of sets, A_1, A_2, \dotsc be given. By definition,
\limsup\limits _{n \to \infty} A_n :=\bigcap\limits _{k =1} ^{\infty} \bigcup\limits _{n =k} ^{\infty} A_n … [1]
We claim
\limsup\limits _{n \to \infty} A_n := \bigcap\limits _{k =1} ^{\infty} \{\omega:\; \omega \in A_n \quad \mathrm {infinitely\; often}\} … [2]
By definition of union,
=\bigcap\limits _{k =1} ^{\infty} \{\omega:\; \exists n,\; n \geq k,\; \omega \in A_n\}
By definition of intersection,
=\{\omega:\; \forall k\; \exists n,\; n \geq k,\; \omega \in A_n\} … [3]
The meaning is completely the same as “infinitely often”, as claimed.

Alternatively, we may define
\limsup\limits _{n \to \infty} A_n =: \{\omega: \limsup \mathbf{1}_{\{A_n\}} [x] =1\}

Similarly, suppose
\liminf\limits _{n \to \infty} A_n :=\bigcup\limits _{k =1} ^{\infty} \bigcap\limits _{n =k} ^{\infty} A_n
We claim
= \bigcap\limits _{k =1} ^{\infty} \{\omega:\; \omega \in A_n \quad \mathrm {for\; all\; but\; finitely\; many}\; n\}
By definition of intersection,
=\bigcup\limits _{k =1} ^{\infty} \{\omega:\; \forall n,\; n \geq k,\; \omega \in A_n\}
By definition of union,
=\{\omega:\; \exists k\; \forall n,\; n \geq k,\; \omega \in A_n\}
The meaning is completely the same as “all but finitely many”, as claimed.

Alternatively, we may define
\liminf\limits _{n \to \infty} A_n =: \{\omega: \liminf \mathbf{1}_{\{A_n\}} [x] =1\}

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