Definition for not divergent

Question

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 $N$?

Answer 1

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 $$ \limsup x_n>-\infty.$$

Answer 2

By definition, $\lim_{n\to\infty} x_n=-\infty$, iff

$\bullet\quad $ for each $M>0$ there is an $n_0$ such that $x_n<-M$ for all $n>n_0$.

Negation of $\ \bullet\ $ means that there is an $M>0$ such that $x_n\geq -M$ for numbers $n$ that can be arbitrarily large, or, in colloquial terms: for infinitely many $n$.

(I took $M>0$ only to enforce the intuition.)