This tutorial discusses the definition of the supremum of a set of numbers. In Part 1, we'll discuss how the definition of the supremum is related to the definition of the maximum. In Part 2, we'll discuss the details of a formal proof involving suprema.
Part 1
It is important to realize that we do not prove anything in this first video. Instead, we're developing intuition for the definition.
After you've completed this tutorial, you might want to reconsider the examples we discussed in the video above. Can you now prove that the supremum of $(0,1)$ is 1?
Part 2
In this part, we'll consider the following problem:
"Determine the supremum of the set $A = \{ \frac{m}{n} : m,n \in \mathbb{N}, m < 2n \}$."
Before watching the video, you should consider the following questions:
- What do you think the supremum of $A$ is? Try to answer even if you aren't sure how to write your proof.
- When you prove the supremum of $A$ is a particular number $s$, how many parts will your proof have? What do you need to argue in each part?
- If you successfully completed the first two questions, try to finish your proof.
After you've watched the video, take a look at your proof once more. How do your proof and the one presented in the film compare? If you weren't able to complete the exercise before watching the film, try again now.