## A proposition with functions

Does the function $$f(x)=\sin x$$ satisfy the following proposition, or its negation? $\forall \varepsilon>0\ \exists K>0\ \forall x: x>K \Rightarrow |f(x)|<\varepsilon.$
The negation of the proposition reads: $$\exists \varepsilon>0\ \forall K>0\ \exists x: x>K \land |f(x)|\ge \varepsilon$$.
This negated proposition is satisfied by choosing $$\varepsilon=\frac12$$ and $$x=\frac\pi2+\lceil K\rceil\pi$$.