For a

Supremum of the Lowest Upper Bound (LUB) of a set is if

1. is an upper bound of
2. is the smallest of such upper bound. That is, if then
   We can also take the contrapositive of 2 and say that every number less than is not an upper bound of .
3. is not an upper bound of or

Similar to the definition of Supremum, we can define infimum or the Greatest Lower Bound (GLB). GLB of a set is if

1. is a lower bound of
2. is the largest of such lowers bounds. That is, if then
   We can also take the contrapositive of 2 and say that greater number less than is not an lower bound of .
3. is not a lower bound of or