Definition:Increasing

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 increasing iff:
 * $\forall x, y \in S: x \; \preceq_1 \; y \implies \phi \left({x}\right) \; \preceq_2 \; \phi \left({y}\right)$

Alternative terms are order-preserving, isotone and non-decreasing.

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

Note
Some sources insist at the point of definition that $\phi$ be an injection for it to be definable as order-preserving, but this is conceptually unnecessary.

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

Thus, let $f$ be a real function.

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

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

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

Also see

 * Strictly increasing
 * Decreasing
 * Monotone