A number is an upper bound of a set if for every we get . Similarly is a lower bound of if for every we get .
If is an upper bound of and additionally it will be called the greatest element of , and such a lower bound will be called the least element of a set. Obviously a set may not have greatest or least element, eg. in there is no least element and neither there is the greatest one.
The least upper bound of a set (if it exists) will be called the supremum of and denoted as . The greatest lower bound of (if it exists) will be called infimum of and denoted . Notice that sometimes supremum or infimum may not be an element of the given set, eg. .
Other example. Let . Then , because it is the greatest element in this set — for every , and . Now, . Indeed, for all we get , because (so is a lower bound of ). Moreover, it is the greatest lower bound, because if , let be a natural number greater than . Then , so is not a lower bound of . We have proved that is its infimum.
If a set has no upper bound we will write and if it has no lower bound, we will write .