Definition for not divergent What is the definition of real numbers $\{x_n\}$ does not diverge to -$\infty$?  
Would it be $x_n$ does not go to infinity if and only if there exist $M>0$ for all $N \in \mathbb{N}$ numbers such that $n$ greater than equal to $N$?

As the definition of "$\{x_n\}$ diverges to $-\infty$" is
$$\forall M\colon \exists N \colon \forall n\ge N\colon x_n<M$$
the negation is
$$\begin{align}\neg\forall M\colon \exists N \colon \forall n\ge N\colon &x_n<M&\iff\\
\exists M\colon \neg\exists N \colon \forall n\ge N\colon &x_n<M&\iff\\
\exists M\colon \forall N \colon \neg\forall n\ge N\colon &x_n<M&\iff\\
\exists M\colon \forall N \colon \exists n\ge N\colon \neg(&x_n<M)&\iff\\
\exists M\colon \forall N \colon \exists n\ge N\colon &x_n\ge M\end{align}$$
or

There exists $M$ such that $x_n\ge M$ infinitely often

or as well
$$ \limsup x_n>-\infty.$$
