Bounded below

Definition

A set is considered bounded below if there exists a real number that serves as a lower limit for the elements in that set. This means that no element in the set is less than this lower limit, providing a boundary that the elements cannot fall below. Understanding this concept is crucial for grasping related ideas like supremum and infimum, as well as recognizing the significance of the greatest lower bound property, which states that every non-empty set of real numbers that is bounded below has a greatest lower bound or infimum.

5 Must Know Facts For Your Next Test

1. A set being bounded below means there is at least one real number that is less than or equal to all elements of the set.
2. If a set has a minimum element, then that minimum element is also its infimum.
3. Not all sets have to be bounded below; for example, the set of all real numbers is not bounded below since it extends infinitely in the negative direction.
4. The existence of a lower bound can impact the convergence of sequences derived from the set, particularly in analysis.
5. In practical applications, understanding whether a function or sequence is bounded below can help determine its behavior and limits.

Review Questions

• How does being bounded below relate to the concept of infimum for a given set?
  • When a set is bounded below, it ensures that there exists at least one real number that acts as a lower limit for the elements in that set. The infimum represents the greatest of these lower bounds. If the set has a smallest element, that element will be both the minimum and the infimum. However, even if a minimum does not exist, as long as the set is bounded below, an infimum will still exist according to the greatest lower bound property.
• Discuss how the greatest lower bound property applies to sets that are bounded below and what implications this has.
  • The greatest lower bound property states that any non-empty subset of real numbers that is bounded below will always have an infimum. This means if we take any collection of numbers where we can identify a boundary beneath them, there will always be a well-defined lowest point they approach but do not necessarily reach. This guarantees completeness in the real numbers and assures us that we can analyze convergence and limits effectively.
• Evaluate how knowing whether a function is bounded below influences its limits and continuity, particularly in real analysis.
  • Understanding if a function is bounded below helps determine its behavior and possible limits as inputs approach certain values or infinity. If a function has a lower bound, it indicates that it will not decrease indefinitely, which can simplify analysis on its limits and continuity. Additionally, if we know a function remains above a certain value across its domain, it can lead to conclusions about convergence in sequences derived from it and support finding critical points when determining maximas or minimas.

Related terms

infimum: The infimum of a set 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.

greatest lower bound property: This property asserts that any non-empty subset of real numbers that is bounded below must have an infimum, ensuring the existence of a greatest lower bound.

bounded above: A set is bounded above if there exists a real number that acts as an upper limit, meaning no element in the set exceeds this upper limit.