Monotone Sequences

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $\le$ be a partial order on a set $X$, with strict relation $<$, and let $(a_{n})$ be a sequence in $X$.

¶ $(a_{n})$ is nondecreasing if $a_{n}\le a_{n+1}$ for every $n\in\mathbb{N}$, strictly increasing if $a_{n}<a_{n+1}$ for every $n\in\mathbb{N}$, nonincreasing if $a_{n+1}\le a_{n}$ for every $n\in\mathbb{N}$, strictly decreasing if $a_{n+1}<a_{n}$ for every $n\in\mathbb{N}$, and monotone if it is nondecreasing or nonincreasing.