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 $$\forall x, y \in S: x \preceq_1 y \iff \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$$.

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

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

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