Definition:Monotone (Order Theory)/Sequence/Real Sequence

Definition

Let $\sequence {x_n}$ be a sequence in $\R$.

Then $\sequence {x_n}$ is monotone if and only if it is either increasing or decreasing.

Also known as

This can also be called a monotonic sequence.

Examples

Example: $\sequence {\paren {-1}^n}$

The first few terms of the real sequence:

$S = \sequence {\paren {-1}^n}_{n \mathop \ge 1}$

are:

$-1, +1, -1, +1, \dotsc$

This is an example of the real sequence:

$S = \sequence {x^n}$

where $x = -1$.

$S$ is not monotone, either increasing or decreasing.

Example: $\sequence {x_n}: x_{n + 1} = \dfrac 2 {x_n + 1}$

Let $\sequence {x_n}$ denote the real sequence defined as:

$x_n = \begin {cases} a : 0 < a < 1 & : n = 1 \\ \dfrac 2 {x_n + 1} & : n > 1 \end {cases}$

Then the subsequences $\sequence {x_{2 n} }$ and $\sequence {x_{2 n + 1} }$ are both monotone:

$\sequence {2 n}$ is strictly decreasing
$\sequence {2 n + 1}$ is strictly increasing

Hence $\sequence {x_n} \to 1$ as $n \to \infty$.