---
tag: 0KN22AAW5R
kind: definition
name: Monotone sequence
msc: [40A05]
defines: 02JBEMG8R4
uses: [0CTV783HYT]
sources:
  - url: https://en.wikipedia.org/wiki/Sequence
    locator: Increasing and decreasing
  - url: https://en.wikipedia.org/wiki/Monotonic_function
    locator: In calculus and analysis
---

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]].
The sequence is increasing if $a_n \le a_{n+1}$ for all $n \in \mathbb{N}$.
The sequence is decreasing if $a_n \ge a_{n+1}$ for all $n \in \mathbb{N}$.
The sequence is monotone if it is increasing or decreasing.
The sequence is strictly increasing if $a_n < a_{n+1}$ for all $n$, and strictly decreasing if $a_n > a_{n+1}$ for all $n$.

## Remarks

Some texts say non-decreasing for increasing in this sense, and they keep the word increasing for strictly increasing.
By induction, $(a_n)$ is increasing if and only if $a_m \le a_n$ for all $m \le n$. This is the definition of a monotone function on $\mathbb{N}$.
The sequence $1/n$ is strictly decreasing. The sequence $(-1)^n$ is not monotone.

## Explanations

### L1 human

A monotone sequence always moves in one direction.
An increasing sequence never goes down, and a decreasing sequence never goes up.