Definition:Strictly 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 strictly increasing if:

$$\forall x, y \in S: x \prec_1 y \iff \phi \left({x}\right) \prec_2 \phi \left({y}\right)$$

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

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

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

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

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

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