bound set of reals

Question

Is $S$ is a bounded, non-empty set ($S \subset \mathbb{R}^n$), then I know it has the least upper bound property. Let $x = \sup S$.

Is $x \in S$? If no, counterexample please. Thank you.

Answer 1

No.

Observe that $1=\sup[0,1)$ while $1\notin[0,1)$.

Answer 2

On real line you can consider the open interval $(0,1)$ where the supremum is $1$ which does not belong to $(0,1)$

For higher dimensions you may define the supernum componentwise in which case the counter example is $(0,1)^n$ That is the Cartesian product of the unit interval in $R^n$