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