# Are strictly increasing/decreasing sequences considered monotone?

A sequence which is either non-decreasing or non-increasing is said to be monotone, this is the definition we have been given. But if a sequence is strictly increasing or strictly decreasing then that doesn't mean it is monotone so monotone is only if it is non-decreasing or non-increasing. Am I correct or not?

A sequence of real numbers $(s_n)$ is:

• monotone if $s_n \leq s_{n+1}$ for all $n \in \mathbb{N}$ or $s_n \geq s_{n+1}$ for all $n \in \mathbb{N}$.
• strictly increasing if $s_n < s_{n+1}$ for all $n \in \mathbb{N}$.
• strictly decreasing if $s_n > s_{n+1}$ for all $n \in \mathbb{N}$.

As $s_n < s_{n+1}$ for all $n \in \mathbb{N}$ implies $s_n \leq s_{n+1}$ for all $n \in \mathbb{N}$, a strictly increasing sequence is monotone.

As $s_n > s_{n+1}$ for all $n \in \mathbb{N}$ implies $s_n \geq s_{n+1}$ for all $n \in \mathbb{N}$, a strictly decreasing sequence is monotone.

In summary, strictly increasing and strictly decreasing sequences are monotone.

A sequence is monotone if it is either non-increasing or non-decreasing.

That is, $(s_n)$ is monotone if either $\forall n \in \mathbb{N}$, $s_{n+1} \geq s_n$ or $\forall n \in \mathbb{N}$, $s_{n+1} \leq s_n$

• @ Ryan Stiles : so if it is strictly increasing or strictly decreasing then it's not monotone? – user134785 May 11 '14 at 10:41

Monotone in words means it can only increase or it can only decrease. And in that case for instance a strictly increasing sequence is monotone since it cannot decrease.