Definition:Strictly Increasing

Ordered Sets
Let $$\left({S; \le_1}\right)$$ and $$\left({T; \le_2}\right)$$ be posets.

Let $$\phi: \left({S; \le_1}\right) \to \left({T; \le_2}\right)$$ be a mapping.

Then $$\phi$$ is strictly increasing if:

$$\forall x, y \in S: x <_1 y \iff \phi \left({x}\right) <_2 \phi \left({y}\right)$$

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 $$\forall n \in \mathbb{N}: x_{n+1} > x_n$$.