# Analysis: Infimum and Supremum

• Rubik
In summary, infimum and supremum are mathematical terms used to describe the smallest and largest possible values in a set of numbers. The infimum is the greatest lower bound and the supremum is the least upper bound. To find these values, one must list out all the numbers in the set and compare them to each other. The values can change depending on the numbers in the set, but will remain the same if the set remains unchanged. These concepts are significant in mathematical analysis as they help define boundaries and play a role in various mathematical concepts such as limits and convergence. Infimum and supremum are related to minimum and maximum, but are not always the same values and instead define the boundaries of the set.
Rubik

## Homework Statement

Find the Supremum and Infimum of S where,
S = {(1/2n) : n is an integer, but not including 0}

## The Attempt at a Solution

Is it right if I got inf{S} = -∞ and sup{S} = ∞

Try checking the values you obtain when substituting n=1,2,...:

{1/2,1/4,1/6,...}

Notice that your values are positive, so that, e.g., -5 is a lower bound. Notice too,

that all the values are smaller than, e.g. 10, so that 10 is an upper bound.

## 1. What is the definition of infimum and supremum?

Infimum and supremum are mathematical terms used to describe the smallest and largest possible values in a set of numbers. The infimum is the greatest lower bound, meaning it is the largest number that is still smaller than all the numbers in the set. The supremum is the least upper bound, meaning it is the smallest number that is still larger than all the numbers in the set.

## 2. How do you find the infimum and supremum of a set of numbers?

To find the infimum and supremum of a set of numbers, you must first list out all the numbers in the set. Then, you can compare the numbers to each other and determine the greatest lower bound (infimum) and the least upper bound (supremum) among them. In some cases, the infimum and supremum may be the same number if there is a maximum or minimum in the set.

## 3. Can the infimum and supremum of a set of numbers change?

Yes, the infimum and supremum of a set of numbers can change depending on the numbers in the set. If even one number is added or removed from the set, it can affect the infimum and supremum. However, if the set remains the same, the infimum and supremum will also remain the same.

## 4. What is the significance of infimum and supremum in mathematical analysis?

Infimum and supremum are important concepts in mathematical analysis because they help define the boundaries of a set of numbers. They also play a crucial role in the study of limits and continuity, as well as in the convergence of sequences and series.

## 5. How are infimum and supremum related to minimum and maximum?

Infimum and supremum are closely related to minimum and maximum, but they are not the same. The minimum and maximum are the smallest and largest values in a set of numbers, respectively. However, the infimum and supremum may or may not be actual numbers in the set, but they still define the boundaries of the set.

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