# Unbounded Monotone Sequence Diverges to Infinity

## Theorem

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

Let $\sequence {x_n}$ be monotone, that is either increasing or decreasing.

### Increasing

Let $\sequence {x_n}$ be increasing and unbounded above.

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

### Decreasing

Let $\sequence {x_n}$ be decreasing and unbounded below.

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