Definition:Increasing

Definition

Ordered Sets

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be posets.

Let $\phi: S \to T$ be a mapping.

Then $\phi$ is increasing iff:

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

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

Real Functions

This definition continues to hold when $S = T = \R$.

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

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

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

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 increasing if

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

Also see

• Strictly increasing
• Decreasing
• Monotone