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

\(\ds x^3\)

\(<\)

\(\ds 8\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(<\)

\(\ds \sqrt [3] 8\)

\(\ds \)

\(=\)

\(\ds 2\)

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 $\sup S \notin S$.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 4$