Infimum

Definition

The infimum, or greatest lower bound, of a set is the largest value that is less than or equal to every element in that set. This concept is critical in understanding limits and bounds of sequences and sets, particularly in the context of completeness, as it helps establish the existence of limits for monotone sequences and plays a key role in analyzing convergence.

5 Must Know Facts For Your Next Test

1. The infimum of a set may not necessarily be an element of that set, meaning it can be a limit point.
2. In the context of monotone sequences, the existence of an infimum guarantees convergence to a limit if the sequence is bounded.
3. The infimum is closely tied to the completeness property of the real numbers, which ensures that all sets with lower bounds have a greatest lower bound.
4. For any set of real numbers, if there exists a least upper bound (supremum), then the infimum provides a complementary perspective by focusing on lower bounds.
5. Calculating the infimum often involves analyzing the limiting behavior of sequences or functions approaching their lower bounds.

Review Questions

• How does the concept of infimum relate to the greatest lower bound property in real analysis?
  • The infimum directly embodies the greatest lower bound property, which asserts that every non-empty subset of real numbers that is bounded below must have an infimum. This means that if you have a set of numbers where you can find a lower limit, the infimum will be that specific greatest lower bound. Understanding this relationship is crucial for proving various properties related to limits and convergence within real analysis.
• Discuss how the infimum impacts monotonic sequences and their convergence.
  • In monotonic sequences, particularly those that are bounded below, the existence of an infimum ensures that the sequence converges to some limit. If a sequence is monotonically increasing and has an infimum, this bound acts as a 'floor' for the values in the sequence, guiding them towards convergence. Hence, recognizing how infimum functions within these sequences can help determine their behavior and stability.
• Evaluate the significance of the infimum in relation to completeness in the real number system.
  • The infimum is significant to the completeness of real numbers because it guarantees that every non-empty set bounded below has a greatest lower bound. This property differentiates real numbers from rational numbers, as there are subsets of rational numbers lacking an infimum. By understanding this completeness criterion through the lens of infimum, one can grasp why certain limits and bounds exist within real analysis and how they support foundational principles like continuity and differentiability.

Related terms

Supremum: The supremum, or least upper bound, of a set is the smallest value that is greater than or equal to every element in that set.

Greatest Lower Bound Property: This property states that every non-empty subset of real numbers that is bounded below has an infimum in the real numbers.

Monotonic Sequence: A sequence that is either non-increasing or non-decreasing throughout its domain, which helps in determining convergence and limits.