Is there a notation for saying that $x_n\geq c$ from some $n$ and on? I am looking for something like $\lim\inf x_n \geq c$, but I need that from some point and on it is $\geq c$, not just that the limit is $\geq c$. 

Perhaps you are looking for a notation like $$\forall \, n \ge N_0  : x_n \ge c$$
where $\forall$ means "for all" and $N_0$ is the "some point and on" chosen by you.
