Monotone Sequence

Definition

Let $(x_n)$ be a sequence of real numbers

Then

• $(x_n)$ is increasing if it satisfies $x_1\le x_2\le \cdots \le x_n\le x_{n+1}\le \dots$
• $(x_n)$ is decreasing if it satisfies $x_1\ge x_2\ge \cdots \ge x_n\ge x_{n+1}\ge \dots$
• $(x_n)$ is monotone if it’s either increasing or decreasing