Unbounded Monotone Sequence Diverges to Infinity/Decreasing

Theorem
Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ be decreasing and unbounded below.

Then $x_n \to -\infty$ as $n \to \infty$.

Proof
Let $H > 0$.

As $\sequence {x_n}$ is unbounded above:
 * $\exists N: x_N > H$

As $\sequence {x_n}$ is decreasing:
 * $\forall n \ge N: x_n \le x_N < H$

It follows from the definition of divergence to $-\infty$ that $x_n \to -\infty$ as $n \to \infty$.