Real numbers/Subset/Supremum/Definition

For a nonempty subset${\displaystyle {}M\subseteq \mathbb {R} }$,anupper bound${\displaystyle {}T}$ of ${\displaystyle {}M}$ is called the supremum of ${\displaystyle {}M}$, if${\displaystyle {}T\leq S}$holds for all upper bounds ${\displaystyle {}S}$ of ${\displaystyle {}M}$.