Say I have the following event $\{\limsup_{n \to \infty} |X_{n}|> \epsilon\}$, and that $P(\{\limsup_{n \to \infty} |X_{n}|> \epsilon\})$, I also realize that
$\{\limsup_{n \to \infty} |X_{n}|> \epsilon\}^{C}=\{\liminf_{n \to \infty} |X_{n}|\leq\epsilon\}$ and subsequently $P(\{\liminf_{n \to \infty} |X_{n}|\leq\epsilon\})=1$
Is my understanding of the statement "$P(\{\liminf_{n \to \infty} |X_{n}|\leq\epsilon\})=1$" correct in that:
$P(\{\liminf_{n \to \infty} |X_{n}|\leq\epsilon\})=1\iff\exists N \in \mathbb N, \forall n \geq N, |X_{n}|\leq\epsilon$ $P-$almost surely.
Am I confusing two separate notations here or are they equivalent?