Question

How do you prove supremum and infimum of a set?

Answer 1

In mathematics, the supremum and infimum of a set are used to describe the maximum and minimum values in the set, respectively. To prove the supremum and infimum of a set, you need to check if the set is bounded above or below, find the upper or lower bound, and prove two conditions about it. For example, if the set S = {1, 2, 3}, the supremum is 4.

Step-by-step explanation:

In mathematics, the supremum (or least upper bound) and infimum (or greatest lower bound) of a set are used to describe the maximum and minimum values in the set, respectively. To prove the supremum and infimum of a set, you need to follow these steps:

1. Check if the set is bounded above or below. If the set has a finite upper or lower bound, then it will have a supremum or infimum, respectively.
2. If the set is bounded above (or below), find an upper (or lower) bound that is greater (or smaller) than all the elements in the set. This will be the supremum (or infimum) of the set.
3. To prove that the found value is indeed the supremum (or infimum), you need to show two things:

For example, let's say we have the set S = {1, 2, 3}. To find the supremum, we check that the set is bounded above, and the number 4 is an upper bound since it is greater than all the elements. Therefore, the supremum of S is 4. To prove this, we need to show that 4 is an upper bound and that there is no lesser upper bound.