I know that $\tan n,n\in\mathbb{Z}_+$ is dense on $\mathbb{R}$, so there's no lower bound for $\tan n$.
But what if add a positive sequence $n^p,p\geq0,n\in\mathbb{Z}_+$ that increased faster than $n\bmod2\pi$ approaching $\dfrac{π}{2}$?
Where is the critical value of $p$?