Finding an upper bound that is not the supremum

In summary, an upper bound is a value that is greater than or equal to all elements in a set. The supremum, also known as the least upper bound, is the smallest upper bound of a set. It is important to find an upper bound that is not the supremum because not all sets have a supremum, and finding an upper bound that is not the supremum can provide a better understanding of the set and its elements. To find an upper bound that is not the supremum, you can use a variety of methods such as trial and error, mathematical formulas, or algorithms, depending on the specific set and its elements. Even though an upper bound that is not the supremum may not be the exact limit of the set
  • #1
ver_mathstats
260
21
Homework Statement
Let S=(0,1), find a real number M which is an upper bound of S, but M is not the supremum of S.
Relevant Equations
S=(0,1)
I just want to see if I did this correctly, the interval (0,1) has 2 as an upper bound but the supremum of S is 1. So M would be equal to 2?

Thank you.
 
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  • #2
Yes, any number greater than 1 would do.
 
  • #3
PeroK said:
Yes, any number greater than 1 would do.
Okay thank you.
 
  • #4
Just to expand a bit. The Supremum is , by definition/construction, the _Least_ Upper Bound. This implies any other upper bound should be greater than the Sup, as @PeroK wrote.
 

What is an upper bound?

An upper bound is a number that is greater than or equal to all the elements in a set. It serves as an upper limit or maximum value for the set.

What is the supremum of a set?

The supremum of a set is the smallest upper bound of the set. It is also known as the least upper bound.

Why is it important to find an upper bound that is not the supremum?

It is important to find an upper bound that is not the supremum because it helps in understanding the behavior and limits of a set. It also allows for more accurate calculations and analysis of the data.

How do you find an upper bound that is not the supremum?

To find an upper bound that is not the supremum, you can use various methods such as the limit of a sequence, the maximum value of a function, or the largest value in a finite set of data.

Can there be more than one upper bound that is not the supremum?

Yes, there can be multiple upper bounds that are not the supremum for a set. This is because there can be multiple numbers that are greater than or equal to all the elements in a set, but not necessarily the smallest upper bound.

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