# Supremum of Subset of Real Numbers/Examples/Example 7

## Example of Supremum of Subset of Real Numbers

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

$S = \set {x \in \R: x^3 < 8}$

$\sup S = 2$

such that $\sup S \notin S$.

## Proof

 $\ds x^3$ $<$ $\ds 8$ $\ds \leadsto \ \$ $\ds x$ $<$ $\ds \sqrt [3] 8$ $\ds$ $=$ $\ds 2$

Hence:

$S = \set {x \in \R: x < 2}$

and it follows that:

$\sup \set {x \in \R: x^3 < 8} = 2$

and it follows that $\sup S \notin S$.

$\blacksquare$