# Definition:Bounded Ordered Set/Real Numbers

This page is about subsets of the real numbers which are bounded. For other uses, see Definition:Bounded.

## Definition

### Definition 1

Let $T \subseteq \R$ be both bounded below and bounded above in $\R$.

Then $T$ is bounded in $\R$.

### Definition 2

Let $T \subseteq \R$ be a subset of $\R$ such that:

$\exists K \in \R: \forall x \in T: \size x \le K$

where $\size x$ denotes the absolute value of $x$.

Then $T$ is bounded in $\R$.

## Unbounded

$T \subseteq \R$ is unbounded (in $\R$) if and only if it is not bounded.

## Examples

### Example 1

The subset $S$ of the real numbers $\R$ defined as:

$S = \set {1, 2, 3}$

is bounded both above and below.

Some upper bounds of $S$ are:

$100, 10, 4, 3$

Some lower bounds of $S$ are:

$-27, 0, 1$

### Example 2

The subset $T$ of the real numbers $\R$ defined as:

$T = \set {x \in \R: 1 \le x \le 2}$

is bounded both above and below.

Some upper bounds of $T$ are:

$100, 10, 4, 2$

Some lower bounds of $T$ are:

$-27, 0, 1$