We say that a subset $A_0$ of a Partially ordered set $A$ is bounded above if there is an element $b$ of $A$ such that $x\leq b$ for every $x \in A_0$; the element $b$ is called an Upper bound for $A_0$. If the set of all upper bounds for $A_0$ has a Smallest element, that element is called the least upper bound, or the Supremum, of $A_0$. It is denoted by $\sup{A_0}$; it may or may not belong to $A_0$. If it does, it is the largest element of $A_0$.

See Bounded above