Monotone Sequences
Why
If the base set of a sequence has a partial order, then we can discuss its relation to the order of sequence.
Definition
A sequence on a partially ordered set is non-decreasing if whenever a first index precedes a second index the element associated with the first index precedes the element associated with the second element. A sequence on a partially ordered set is increasing if it is non-decreasing and no two elements are the same.
A sequence on a partially ordered set is non-increasing if whenever a first index precedes a second index the element associated with the first index succedes the element associated with the second element. A sequence on a partially ordered set is decreasing if it is non-increasing and no two elements are the same.
A sequence on a partially ordered set is monotone if it is non-decreasing, or non-increasing. An increasing sequence is non-decreasing. A decreasing sequences is non-increasing. A sequence on a partially ordered set is strictly monotone if it is decreasing, or increasing.
Notation
Let $A$ a non-empty set with partial order $\preceq$. Let $\seq{a}$ a sequence in $A$.
The sequence is non-decreasing if $n \leq m \implies a_n \preceq a_m$, and increasing if $n < m \implies a_n \prec a_m$. The sequence is non-increasing if $n \leq m \implies a_n \succeq a_m$, and decreasing if $n < m \implies a_n > a_m$.