Supremum

Definition

The supremum, often referred to as the least upper bound, is the smallest value that is greater than or equal to every number in a given set. This concept is crucial in understanding how we can find bounds for sets of numbers, particularly in the context of real numbers, where every non-empty set that is bounded above has a supremum. It highlights the distinction between the maximum value of a set and its supremum, especially when the maximum does not exist within the set itself.

5 Must Know Facts For Your Next Test

1. The supremum does not need to be an element of the set; it simply needs to be greater than or equal to all elements in the set.
2. For example, the supremum of the open interval (0, 1) is 1, even though 1 is not included in the interval.
3. In finite sets, the supremum is equal to the maximum if the maximum exists within the set.
4. If a set is unbounded above, it does not have a supremum in the real numbers.
5. The concept of supremum is essential in calculus and analysis, as it helps define limits and continuity.

Review Questions

• How does the concept of supremum differ from that of maximum in a given set?
  • The supremum of a set refers to the least upper bound, which may or may not be an element of the set, while the maximum is the largest actual element within that set. For instance, in the set {2, 3, 5}, both the maximum and supremum are 5. However, for the open interval (0, 1), the supremum is 1, but there is no maximum since 1 is not part of this interval.
• Discuss why every non-empty bounded above subset of real numbers has a supremum and how this relates to the completeness axiom.
  • Every non-empty subset of real numbers that is bounded above has a supremum due to the completeness axiom. This axiom ensures that there are no 'gaps' in the real numbers. It guarantees that even if a set doesn't contain its maximum element, there still exists a least upper bound that can serve as its supremum. This property is fundamental to analysis as it provides a solid foundation for further mathematical concepts like limits and convergence.
• Evaluate how the properties of supremum can be applied to analyze sequences and functions in real analysis.
  • In real analysis, understanding supremum allows us to effectively analyze convergence and boundedness of sequences and functions. By applying the concept of supremum to a sequence, we can determine whether it converges or diverges by examining its upper bounds. For instance, if we can find a supremum for a sequence approaching a limit, we can infer properties about its behavior and establish criteria for convergence. This understanding also aids in optimizing functions by locating maxima and minima within intervals.

Related terms

Infimum: The infimum is the greatest lower bound of a set, which is the largest value that is less than or equal to every number in the set.

Bounded Set: A bounded set is a set of numbers that has both an upper and a lower bound, meaning there are specific values that all elements of the set fall between.

Completeness Axiom: The completeness axiom states that every non-empty subset of real numbers that is bounded above has a supremum in the real numbers, emphasizing their structured nature.