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 $\geq c$. 
 A: You can use the word eventually to describe this scenario; for example, you would say that the sequence $\{a_n\}_n$ is eventually bounded below by $c$.
A: As @user notes, the usual term for this notion is "eventually" (introduced or anyway popularized by Kelley in his book General Topology). It's a quantifier, sometimes written $\forall^*$. For any predicate $P(n)$ of integers, $P(n)$ holds eventually (with respect to $n$) iff $(\forall^* n)\, P(n) := (\exists N)(\forall n\ge N)\, P(n)$.
If you need to use the notion a lot, you could make up a notation, such as
$$
(x_n)_{n\in\Bbb N} \le^* (y_n)_{n\in\Bbb N} := (\forall^* n\in \Bbb N)\, x_n \le y_n.
$$
with the understanding that a real $c$ used as a sequence, as in $(y_n)\, ^*{\ge}\, c$, stands for the constantly-$c$ sequence. Note that $\le^*$ is a preorder — it's reflexive and transitive.
Just as an aside, the dual notion to "eventually" is "frequently" (again, Kelley's term): $P(n)$ holds frequently (w.r.t. $n$) means $(\forall N)(\exists n\ge N)\,P(n)$. Over the integers, it's equivalent to "$P(n)$ holds infinitely often". If we write $\exists^*$ for "frequently", then these two quantifiers obey the familiar rules: $\forall^* = \lnot \exists^*\lnot$ and $\exists^* = \lnot \forall^*\lnot$, as is easily verified:
$$\begin{align}
\lnot (\exists^* n)\lnot P(n)&\iff \lnot (\forall N)(\exists n\ge N)\lnot P(n) \\
&\iff (\exists N)(\forall n\ge N)\lnot\lnot P(n) \\
&\iff (\forall^* n) P(n). \\
\end{align}$$
A: 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.
A: I think I've seen things like "$x_n \ge c$ for all $n \gg 1$", but using words is definitely clearer.
A: The phrase "for all sufficiently large $n$" works also. If you want more detail (not necessary in my opinion) you can say that there exists $N$ such that $x_n\ge c$ for all $n\ge N$. 
