Bounded above

Definition

A set of numbers is said to be bounded above if there exists a real number that is greater than or equal to every number in the set. This means that all elements of the set do not exceed a certain limit, making it essential in understanding the behavior of sequences and functions. Boundedness leads to the concepts of supremum and infimum, which help in identifying the least upper bound of a set and understanding the properties of real numbers, intervals, and their absolute values.

5 Must Know Facts For Your Next Test

1. A set can be bounded above even if it does not include its supremum.
2. If a set is bounded above, then it must have a supremum, according to the least upper bound property.
3. Boundedness can be visualized on a number line where all elements fall within a certain segment defined by the upper limit.
4. The concept of being bounded above applies not only to finite sets but also to infinite sets like converging sequences.
5. Knowing whether a set is bounded above is crucial for applying various mathematical theorems and principles related to limits and continuity.

Review Questions

• How does the concept of being bounded above relate to the existence of a supremum for a given set?
  • A set being bounded above guarantees that there is at least one real number that serves as an upper limit for its elements. This leads directly to the existence of a supremum, which is defined as the least upper bound. If a set has an upper bound, then it must have a supremum, which could either be a member of the set or an external value that bounds the set from above.
• In what ways do intervals demonstrate the idea of being bounded above, and how does this impact their properties?
  • Intervals can be classified as bounded or unbounded based on their endpoints. A bounded interval has both upper and lower limits, which means it is bounded above and below. This characteristic impacts their properties significantly; for instance, all bounded intervals contain both their supremum and infimum if they are closed. Understanding how intervals behave concerning being bounded helps in analyzing functions defined over these intervals.
• Evaluate the implications of a sequence being bounded above in terms of convergence and its behavior as n approaches infinity.
  • When a sequence is bounded above, it implies that its terms will not exceed a certain value as n approaches infinity. This boundedness is crucial in determining whether the sequence converges or diverges. If a sequence is also monotonic (either increasing or decreasing), being bounded above guarantees that it converges to its supremum. Thus, the relationship between being bounded above and convergence provides insight into the long-term behavior of sequences.

Related terms

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

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

Interval: A range of real numbers between two endpoints, which can be open, closed, or half-open, used to describe bounded sets.