Definition:Increasing

[edit] 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 $φ$ is increasing iff $\forall x, y \in S: x \preceq_1 y \iff \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$ .

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

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

Thus, let $f$ be a real function.

Then $f$ is increasing iff $x \le y \iff f \left({x}\right) \le f \left({y}\right)$ .

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

Then $\left \langle {x_n} \right \rangle$ is increasing if $\forall n \in \N: x_n \le x_{n+1}$ .