$ \lim \sup_{n \to \infty} n \mathbb P \left ( \sup_{t \in [ 0 , T ]} S_t > n \right ) > 0 $ means $S_t$ is unbounded above almost surely.

Question

In a text book, the author states that $$ \lim \sup_{n \to \infty} n \mathbb P \left ( \sup_{t \in [ 0 , T ]} S_t > n \right ) > 0 $$ means that $S_t$ is unbounded above almost surely. Can someone give some details please about that ?

It seems to me that being unbounded (above) $\iff $ $$ \forall n \in \mathbb R, \mathbb P( S_t > n ) > 0 $$

Are those statements really equivalent ?

Also, in the same way, one could read in the same text book that equation :

$$ \lim_{n \to \infty} n \mathbb P \left ( \inf_{t \in [ 0 , T ]} Z_t < - n \right ) = 0 $$ What is the interpretation of that equation ? That the process $Z_t$ is bounded below ?