Definition:Monotone (Order Theory)

Ordered Sets
Let $$\left({S; \preceq_1}\right)$$ and $$\left({T; \preceq_2}\right)$$ be posets.

Let $$\phi: \left({S; \preceq_1}\right) \to \left({T; \preceq_2}\right)$$ be a mapping.

Then $$\phi$$ is monotone iff it is either increasing or decreasing.

Note that this definition also holds if $$S = T$$.

Real Functions
This definition continues to hold when $$S = T = \R$$.

Thus, let $$f$$ be a real function.

Then $$f$$ is monotone iff it is either increasing or decreasing.

Sequences
Let $$\left \langle {x_n} \right \rangle$$ be a sequence in $\R$.

Then $$\left \langle {x_n} \right \rangle$$ is monotone if it is either increasing or decreasing.