We say that a subset $A_0$ of a Partially ordered set $A$ is bounded below if there is an element $a$ of $A$ such that $a\leq x$ for every $x \in A_0$; the element $a$ is called a Lower bound for $A_0$. If the set of all upper bounds for $A_0$ has a Largest element, that element is called the greatest upper bound, or the Infimum, of $A_0$. It is denoted by $\inf{A_0}$; it may or may not belong to $A_0$. If it does, it is the smallest element of $A_0$.