# Least Upper Bound: What Is It & How to Prove It

• soul
In summary, the concept of "least upper bound" refers to the smallest number that is greater than every element in a set. In the case of the sequence {an}, it would be the smallest number that is greater than every term in the sequence. This term is often misunderstood as simply being an upper bound, but it must also be the smallest such upper bound. This can be proven by looking at examples, such as the set (0,1), where 1 is the least upper bound. Additionally, in some cases, the least upper bound may be 0, as seen in the case of negative real numbers.
soul
I am confused about the concept of "least upper bound". Is this line the limt of the {an} sequence. If so, how can we prove it?

the least upper bound of a set E is a number greater than every element of E s.t. if another number is less than the least upper bound, it is in E.

The term least upper bound is a loaded term.
First of all it is an upper bound of the sequence {an}, meaning it is greater than every term of the sequence. More generally the upper bound of some set is some number greater than any number in the set.
Secondly it is the smallest such upper bound. So if A is a least upper bound for {an} and B is some other upper bound then A<B

matticus said:
the least upper bound of a set E is a number greater than every element of E s.t. if another number is less than the least upper bound, it is in E.

This is false, consider the set (0,1) then 1 is obviously the least upper bound, and also 0 is less than 1, but 0 is not in the set.

With a search on google and with your help I understood the point. Thanks for your help guys.

Think of an upper bound of nevative real numbers. Obviously 0 and any number greater than 0 is an upper bound. But the least is 0.

## 1. What is a least upper bound?

A least upper bound, also known as a supremum, is the smallest number that is greater than or equal to all the numbers in a given set. It is a fundamental concept in mathematics and is often used to prove the existence of certain numbers or to establish the convergence of a sequence.

## 2. How is a least upper bound different from a maximum?

A maximum is the largest number in a given set, while a least upper bound is the smallest number that is greater than or equal to all the numbers in the set. In other words, a maximum must be an element of the set, while a least upper bound can be a limit point or a number that is not in the set.

## 3. How do you prove the existence of a least upper bound?

To prove the existence of a least upper bound for a set of real numbers, you need to show that the set has an upper bound and that there is no smaller number that is also an upper bound. This can be done using the least upper bound axiom, which states that every non-empty set of real numbers that is bounded above has a least upper bound.

## 4. What are some common examples of least upper bounds?

One common example of a least upper bound is the number π, which is the least upper bound of the set of rational numbers that are less than or equal to π. Another example is the number √2, which is the least upper bound of the set of rational numbers that are less than or equal to √2.

## 5. How is the concept of least upper bound used in real life?

The concept of least upper bound is used in various fields such as economics, engineering, and computer science. For example, in economics, the least upper bound is used to determine the maximum price that consumers are willing to pay for a product. In engineering, it is used to establish the maximum load that a structure can handle. In computer science, it is used in algorithms to find the shortest path between two points.

• Calculus
Replies
1
Views
1K
• Calculus
Replies
5
Views
1K
• Calculus
Replies
1
Views
982
• Calculus
Replies
4
Views
1K
• Calculus
Replies
1
Views
527
• Calculus
Replies
3
Views
3K
• Calculus
Replies
5
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
3
Views
4K
• Set Theory, Logic, Probability, Statistics
Replies
17
Views
1K
• Calculus
Replies
9
Views
1K