Definition:Unbounded Divergent Sequence/Real Sequence/Negative Infinity

Definition
Let $\sequence {x_n}$ be a sequence in $\R$.

$\sequence {x_n}$ diverges to $-\infty$ :
 * $\forall H \in \R_{>0}: \exists N: \forall n > N: x_n < -H$

That is, whatever positive real number $H$ you choose, for sufficiently large $n$, $x_n$ will be less than $-H$.

We write:
 * $x_n \to -\infty$ as $n \to \infty$

or:
 * $\displaystyle \lim_{n \mathop \to \infty} x_n \to -\infty$

Also known as
The statement:
 * $\sequence {x_n}$ diverges to $-\infty$

can also be stated:
 * $\sequence {x_n}$ tends to $-\infty$
 * $\sequence {x_n}$ is unbounded below.

Also see

 * Definition:Divergent Real Sequence to Positive Infinity
 * Definition:Unbounded Divergent Real Sequence