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

admits a supremum:


 * $\sup S = 2$

such that $\sup S \notin S$.

Proof
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 $\sup S \notin S$.