$$X \subset \mathbb{R}$$
$$\exists c \in \mathbb{R} \; \text{such that}$$
$$x \lt c, \forall x \in X$$


$$sup(\langle 0, 1 \rangle) = 1 \notin \langle 0, 1 \rangle$$
$$sup(\langle 0, 1 ] \,)$$


$$sup(\mathbb{R}) = \infty$$