Background to my question:
Given that $(X_{n})_{n}$ are random variables on $(\Omega, \mathcal{F}, P)$ and for $\epsilon > 0$: $\sum_{n \in \mathbb N}P(|X_{n}|>\epsilon)<\infty$
It follows from BCL that $P(\limsup_{n \to \infty}|X_{n}|>\epsilon)=0$ and therefore $P(\liminf_{n \to \infty}|X_{n}|\leq \epsilon)=1$
As the title suggests, is the definition of $P(\liminf_{n \to \infty}|X_{n}|\leq \epsilon)=1$ simply equivalent to:
$\exists N \in \mathbb N: |X_{n}|\leq\epsilon, \forall n \geq N, P-$a.s. or are there differences between the two?
By definition, \begin{align*} \omega \in \liminf A_n &\Longleftrightarrow \exists m_0 \in \mathbb N : \omega \in \bigcap_{n\geq m_0} A_n\\ &\Longleftrightarrow \exists m_0 \in \mathbb N \, \forall n\geq m_0 : \omega \in A_n \\[.5em] &\Longleftrightarrow \omega \in A_n \text{ taken from (countably) infinitely many $n \in \mathbb N$ }\\ &\Longleftrightarrow \omega \in A_n \text{ for all except finitely many $n \in \mathbb N$ (i.e., for cofinitely many $n$) } \\ &\Longleftrightarrow \omega \text{ in ultimately all of $A_n$ $(n \in \mathbb N)$ } \\ &\Longleftrightarrow A_n \text{ ultimately (ult.) } \\[.5em] \end{align*} and \begin{align*} \omega \in \limsup A_n &\Longleftrightarrow \forall m \in \mathbb N : \omega \in \bigcup_{n\geq m} A_n \\ &\Longleftrightarrow \forall m \in \mathbb N \, \exists n_0(m)\geq m : \omega \in A_{n_0(m)} \\[.5em] &\Longleftrightarrow \omega \text{ appears in infinitely many of the $A_n$ }\\ &\Longleftrightarrow \omega \in A_n \text{ for all but finitely many $n \in \mathbb N$ (i.e., cofinitely many $n$) } \\ &\Longleftrightarrow A_n \text{ for infinitely many $n \in \mathbb N$ } \\ &\Longleftrightarrow A_n \text{ infinitely often (i.o.)}. \\[.5em] \end{align*} So if $\mathbb P(\limsup A_n) = 0$, then the probability that infinitely many of $A_n$ occur is 0.