Say a sequence $\left(a_n\right)_{n\in \mathbb{N}}$ is frequent in some set $A$ if $\forall N \in \mathbb{N} : \exists n \ge N : a_n \in A$.
Would the negation of this be $\exists N \in \mathbb{N} : \forall n \ge N : a_n \notin A$?
So, there's some natural for which the sequence after this natural (as the sequence index) contains no elements in the set?
I am trying to prove by contradiction certain properties of a set with infinitely many of some element.
Yes, it's correct. For the negation of a whole statement you negate each quantifier and the property at the very end.