Completeness Property of R: The completeness property states that any non-empty set of real numbers that is not empty itself must contain a least one real n...
Completeness Property of R:
The completeness property states that any non-empty set of real numbers that is not empty itself must contain a least one real number. In simpler words, if you have a set of real numbers that is not empty, and if you remove all the elements from the set that are not real, the resulting set still contains at least one real number.
Examples:
Consider the set of real numbers between 0 and 1, [0, 1). This set is not empty, but it contains no real numbers. However, the set does contain the real number 0.
Consider the set of all rational numbers, Q. This set is also not empty, but it contains no real numbers. However, the set does contain the real number 0.
Consider the set of all real numbers that are greater than 1, (1, ∞). This set is not empty, but it contains no real numbers. However, the set does contain the real number 2.
The completeness property is a fundamental property of the real number system. It means that the real number system is complete, meaning that any non-empty set of real numbers must contain at least one real number