What is the Least Upper Bound property of real numbers?
Understand the Problem
The question is asking for an explanation of the Least Upper Bound property (also known as the Completeness Axiom) of real numbers, which states that every non-empty set of real numbers that is bounded above has a least upper bound (supremum) in the real numbers.
Answer
Any non-empty set of real numbers with an upper bound has a least upper bound.
The Least Upper Bound property states that any non-empty set of real numbers that is bounded above has a least upper bound, also known as the supremum, in real numbers.
Answer for screen readers
The Least Upper Bound property states that any non-empty set of real numbers that is bounded above has a least upper bound, also known as the supremum, in real numbers.
More Information
The Least Upper Bound property is essential for real numbers, allowing for the development of calculus due to its completeness, unlike rational numbers which do not have this property.
Tips
A common mistake is confusing the least upper bound property with the greatest lower bound property. Remember, the least upper bound (or supremum) is the smallest number that is greater than every element in the set.
Sources
- The Least Upper Bound Property - Wikipedia - en.wikipedia.org
- 8.5: The Least Upper Bound Property - Mathematics LibreTexts - math.libretexts.org
- Least Upper Bound Axiom - math.hws.edu
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