Infimum

Definition

The infimum of a set of real numbers is the greatest lower bound of that set, meaning it is the largest number that is less than or equal to every number in the set. This concept is essential in understanding how numbers are ordered and their density within the real number line. The infimum can be an element of the set, or it may not be; if it's not, it signifies that there is no smallest number in the set.

5 Must Know Facts For Your Next Test

1. The infimum exists for any non-empty set of real numbers that is bounded below.
2. If the infimum is a member of the set, it is also the minimum value of that set.
3. For any given set, if the infimum does not belong to the set, there exists no smallest element in that set.
4. In mathematical notation, if A is a set, then inf(A) refers to its infimum.
5. The concept of infimum is crucial in calculus and analysis, especially when dealing with limits and continuity.

Review Questions

• How does the infimum relate to other bounds of a set of real numbers?
  • The infimum serves as the greatest lower bound for a set, which means it is higher than or equal to every element in that set. In contrast, the supremum acts as the least upper bound. Understanding this relationship helps clarify how sets can be contained within bounds and how their limits behave under various mathematical operations.
• Discuss how knowing the infimum of a bounded set can aid in understanding its structure and properties.
  • Knowing the infimum provides insights into how a bounded set behaves at its lower end. It reveals whether there are gaps or whether the elements cluster close together near this lower boundary. This information can lead to further analysis about limits, continuity, and the overall distribution of elements within the set.
• Evaluate how the concept of infimum might apply when analyzing sequences that converge towards a limit.
  • When analyzing converging sequences, understanding their infimum helps determine their behavior as they approach their limit. If a sequence is bounded below and converges to a limit, its infimum indicates the lowest value it approaches without actually being reached if it's not included in the sequence. This application illustrates how limits relate to ordered structures and density within real numbers, highlighting crucial properties like completeness in analysis.

Related terms

Supremum:

The supremum of a set is the least upper bound, which means it is the smallest number that is greater 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 real numbers that are greater than all elements and less than all elements in the set.

Density: Density refers to how closely packed the elements of a set are within a given interval on the real number line, and whether there are gaps between these elements.