Infimum and supremum

Summary

In mathematics, the infimum (abbreviated inf; : infima) of a subset $S$ of a partially ordered set $P$ is the greatest element in $P$ that is less than or equal to each element of $S,$ if such an element exists. If the infimum of $S$ exists, it is unique, and if b is a lower bound of $S$, then b is less than or equal to the infimum of $S$. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. The supremum (abbreviated sup; : suprema) of a subset $S$ of a partially ordered set $P$ is the least element in $P$ that is greater than or equal to each element of $S,$ if such an element exists. If the supremum of $S$ exists, it is unique, and if b is an upper bound of $S$, then the supremum of $S$ is less than or equal to b. Consequently, the supremum is also referred to as the least upper bound (or LUB).