Supremum The supremum of a set of real numbers is the least upper bound of that set. In other words, it is the smallest real number that is greater than or...
Supremum
The supremum of a set of real numbers is the least upper bound of that set. In other words, it is the smallest real number that is greater than or equal to all real numbers in the set.
Infimum
The infimum of a set of real numbers is the largest lower bound of that set. In other words, it is the largest real number that is less than or equal to all real numbers in the set.
For example, consider the set of real numbers between 0 and 1, which can be represented by the interval [0, 1). The supremum of this set would be 1, since it is the smallest real number that is greater than or equal to all other real numbers in the set. The infimum of this set would be 0, since it is the largest real number that is less than or equal to all other real numbers in the set.
The supremum and infimum of a set of real numbers are both important concepts in real analysis. They are used to study the maximum and minimum values of functions, and to determine the existence of limits