Unbounded Monotone Sequence Diverges to Infinity

Theorem
Let $$\left \langle {x_n} \right \rangle$$ be a sequence in $\mathbb{R}$.

Let $$\left \langle {x_n} \right \rangle$$ be monotone, i.e. either increasing or decreasing.

Increasing
Let $$\left \langle {x_n} \right \rangle$$ be increasing and unbounded above.

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

Decreasing
Let $$\left \langle {x_n} \right \rangle$$ be decreasing and unbounded below.

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

Proof

 * Let $$\left \langle {x_n} \right \rangle$$ be increasing and unbounded above.

Let $$H > 0$$.

As $$\left \langle {x_n} \right \rangle$$ is unbounded above, $$\exists N: x_N > H$$.

As $$\left \langle {x_n} \right \rangle$$ is increasing, then $$\forall n \ge N: x_n \ge x_N > H$$.

It follows from the definition of divergent to infinity that $$x_n \to \infty$$ as $$n \to \infty$$.


 * The same argument can be used for the case where $$\left \langle {x_n} \right \rangle$$ is decreasing and unbounded below.