# What does it mean for a sequence to be monotone?

Dec 31, 2015

It means that the sequence is always either increasing or decreasing, it the terms of the sequence are getting either bigger or smaller all the time, for all values bigger than or smaller than a certain value.

#### Explanation:

Here is the precise definitions :

• A sequence $\left({x}_{n}\right) \in \mathbb{R} \mathmr{and} \mathbb{C} , n \in \mathbb{N}$ is called monotone increasing $\iff \exists k \in \mathbb{N}$such that ${x}_{n + 1} \ge {x}_{n} \forall n \ge k$.
• A sequence $\left({x}_{n}\right) \in \mathbb{R} \mathmr{and} \mathbb{C} , n \in \mathbb{N}$ is called monotone decreasing $\iff \exists k \in \mathbb{N}$such that ${x}_{n + 1} \le {x}_{n} \forall n \ge k$.

Note also that $\left({x}_{n}\right)$ is said to be bounded $\iff \exists M \in \mathbb{N}$such that $\left({x}_{n}\right) \le M \forall n \in \mathbb{N}$.

In addition, $\left({x}_{n}\right)$ converges to a limit $x \in \mathbb{R} \mathmr{and} \mathbb{C} \iff \forall \epsilon > 0 \exists N \in \mathbb{N} > 0$ such that $| {x}_{n} - x | < \epsilon \forall n > N$.

Furthermore, there is a theorem which states that every bounded, momotonic sequence is convergent.