Is it true that for a real function f: $$ \limsup_{t \to \infty} f(t) = \sup_{(t_n)_{n \in \mathbb{N}}, (t_n) \to \infty}\limsup_{t_n \to \infty} f(t_n) $$ ?
Where the first limsup is seen in the continuous sense and the second in a discrete sense.
I'm interested in this kind of properties as I'm working on some Brownian Motion proofs where I need easy ways for manipulating both kinds of limsup. I'm not directly interested in the proof of the result but rather on it's truthfulness. Still any intuition or sketch of proof is welcome...
What useful characterizations allow one to work and move between both kinds of limsup without much trouble?