Definition:Supremum of Set/Real Numbers

Definition
Let $T \subseteq \R$.

A real number $c \in \R$ is the supremum of $T$ in $\R$ iff:


 * $(1): \quad c$ is an upper bound of $T$ in $\R$
 * $(2): \quad c \le d$ for all upper bounds $d$ of $T$ in $\R$.

The supremum of $T$ is denoted $\sup T$.

If there exists a supremum of $T$ (in $\R$), we say that:
 * $T$ admits a supremum (in $\R$) or
 * $T$ has a supremum (in $\R$).

Also see

 * Definition:Infimum of Subset of Real Numbers


 * Supremum and Infimum are Unique