- #1
- 3
- 0
please i need your help!
prove: "A nonempty set of real numbers bounded from below has a greatest lower bound."
prove: "A nonempty set of real numbers bounded from below has a greatest lower bound."
HallsofIvy said:Mariouma and Cristo are assuming that you are allowed to use the fact that "if a set of real numbers has an upper bound then it has a least upper bound"- what Cristo called the "completeness axiom". Is that true?
adityab88 is, I think, assuming that you have to prove that completenss axiom from the definition of the real numbers. It very easy to do that from the Dedekind cut definition of the real numbers.
The Greatest Lower Bound, also known as the infimum, is the largest number that is less than or equal to all the elements in a given set. It is denoted as inf(S) or GLB(S).
Proving the Greatest Lower Bound is important because it helps to establish the minimum value in a set, which is useful in understanding the properties and behavior of the set. It also aids in solving problems related to optimization and analysis.
The proof of the Greatest Lower Bound involves showing that the infimum is a lower bound for the set and that it is the greatest lower bound by demonstrating that any other lower bound must be greater than or equal to the infimum.
Greatest Lower Bound and Least Upper Bound are dual concepts, with the former representing the smallest value in a set and the latter representing the largest value. While GLB is the largest number that is less than or equal to all the elements in a set, LUB is the smallest number that is greater than or equal to all the elements in a set.
The Greatest Lower Bound has various applications in fields such as economics, finance, and engineering. It is used in optimization problems, stock market analysis, and designing efficient systems. It is also used in measuring the performance of algorithms and data structures.