# Finding the Infimum and Supremum

• MHB
• slwarrior64
In summary, the conversation discusses the concept of infimum and supremum and how to prove a question related to it. The speaker mentions their struggle with understanding the formal proof style and asks for clarification. The other person explains that the supremum does not exist because the set has no upper bound and the infimum is 0. The conversation also touches upon inequalities and how they relate to the proof.
slwarrior64

Hello, I feel like I am struggling with this more than I should. I can tell intuitively what the infimum and supremum are, but I am pretty sure that I need a more formal proof style answer. How would one actually prove this question?

One doesn't "prove" a questiom! One simply answers a question and then, prehaps, proves that the ansser is correct. You say "I can tell intuitively what the infimum and supremum are," Okay, what are they?

I got that the Supremum does not exist because the set has no upper bound, and I got that the lower bound was 0 because once k reaches the negatives you are geting a negative exponent which becomes a fraction with an increasingly large denominator. I was unsure about what to do with it because the other problems we had done discussed the inf and sup of things that included inequalities so we had to rearrange the inequality and prove it true.

if you want inequalities, you know that $$2^k> 0$$ for all k. You also can say that, given any positive integer, N, there exist a positive integer k so that $$2^k>N$$. That certainly shows that $$2^k$$ has no upper bound. But it is also true that $$2^{-k}< \frac{1}{N}$$ showing that the infimum is 0.

Okay, I think I get what you are saying, but there are some weird formatting things in your reply that I don't really understand:
if you want inequalities, you know that $$2^k> 0$$ for all k. You also can say that, given any positive integer, N, there exist a positive integer k so that $$2^k>N$$. That certainly shows that $$2^k$$ has no upper bound. But it is also true that $$2^{-k}< \frac{1}{N}$$ showing that the infimum is 0.

## What is the infimum and supremum?

The infimum and supremum are mathematical concepts used to describe the smallest and largest values in a set of numbers, respectively. The infimum is the greatest lower bound, meaning it is the largest value that is less than or equal to all other values in the set. The supremum is the least upper bound, meaning it is the smallest value that is greater than or equal to all other values in the set.

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

To find the infimum and supremum, you must first arrange the numbers in the set in ascending order. The infimum will be the first number in the set, and the supremum will be the last number in the set.

## What is the difference between the infimum and supremum?

The main difference between the infimum and supremum is that the infimum is the largest value that is less than or equal to all other values in the set, while the supremum is the smallest value that is greater than or equal to all other values in the set.

## Can a set have multiple infimums or supremums?

No, a set can only have one infimum and one supremum. However, a set may not have an infimum or supremum if the set is unbounded or if the infimum or supremum does not exist.

## Why is finding the infimum and supremum important?

Finding the infimum and supremum is important because it helps us understand the boundaries and limits of a set of numbers. It is also useful in many mathematical concepts and applications, such as optimization and analysis.

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