Unbounded Monotone Sequence Diverges to Infinity

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

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.