The supremum of a set is the smallest real number that is greater than or equal to every number in the set.
Two definitions are used for the supremum. The first definition is used before we have established that there cannot be more than one supremum. The first definition allows us to consider the possibility of there being multiple supremums of set, so that we can then prove that this is impossible. With this fact established, the second definition is then introduced. We can show that the two definitions are equivalent.