Definition:Strictly Increasing

Definition

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 $S = T = \R$.

Thus, let $f$ be a real function.

Then $f$ is strictly increasing iff:

$x < y \iff f \left({x}\right) < f \left({y}\right)$

Sequences

Let $\left({S, \preceq}\right)$ be a totally ordered set.

Then a sequence $\left \langle {a_k} \right \rangle_{k \in A}$ of terms of $S$ is strictly increasing iff:

$\forall j, k \in A: j < k \implies a_j \prec a_k$

Real Sequences

The above definition for sequences is usually applied to real number sequences.

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

Then $\left \langle {x_n} \right \rangle$ is strictly increasing if

$\forall n \in \N: x_n < x_{n+1}$

Also see

• Increasing
• Strictly Decreasing
• Strictly Monotone