# Greatest Lower Bound: Prove It!

• mariouma
In summary, the conversation discusses the definition and properties of the greatest lower bound of a nonempty set of real numbers bounded from below. The participants also mention the completeness axiom and its relation to finding the greatest lower bound. The use of Dedekind cuts or Cauchy sequences is suggested to prove the statement.

#### mariouma

prove: "A nonempty set of real numbers bounded from below has a greatest lower bound."

This looks like homework. What thoughts do you have on the question? What is the definition of the greatest lower bound of a set?

A lower bound of a non-empty subset A of R is an element d in R with d <= a for all a A.
An element m in R is a greatest lower bound or infimum of A if
m is a lower bound of A and if d is an upper bound of A then m >= d.

Last edited:
Ok, so here A is bounded below, so this tells you that there exists a lower bound to A. It may now be helpful to consider the set
-A:={-x:x∈A}, and use the completeness axiom to find the least upper bound on -A. The relationship between A and -A should help you find the greatest lower bound of A.

this is not number theory

this is not number theory, but it is instead mathematical analysis.

it is a very basic result that has been used to construct the real numbers and i think you will find it in any standard intro to analysis textbook (i personally recommend principles of mathematical analysis by Walter Rudin).

hope it helps

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.

dedekind cuts/ cauchy sequences

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.

precisely. dedekind cuts or even cauchy sequences can be used prove such a statement.

## What is the definition of Greatest Lower Bound?

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).

## Why is it important to prove the Greatest Lower Bound?

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.

## How do you prove the Greatest Lower Bound?

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.

## What is the difference between Greatest Lower Bound and Least Upper Bound?

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.

## What are some real-life applications of the Greatest Lower Bound?

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.

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