Definition:Unbounded Divergent Sequence/Real Sequence/Negative Infinity

Definition

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

$\sequence {x_n}$ diverges to $-\infty$ if and only if:

$\forall H \in \R_{>0}: \exists N: \forall n > N: x_n < -H$

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:

$\ds \lim_{n \mathop \to \infty} x_n \to -\infty$

The statement:

$\sequence {x_n}$ diverges to $-\infty$

can also be stated:

$\sequence {x_n}$ tends to $-\infty$

$\sequence {x_n}$ is unbounded below.

Let $\sequence {a_n}_{n \mathop \ge 1}$ be the real sequence defined as:

$a_n = -\sqrt n$

Then $\sequence {a_n}$ is divergent to $-\infty$.

1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.26$: Divergent Sequences