Epsilon Definition of Supremum and Infimum | Real Analysis

We prove an equivalent epsilon definition for the supremum and infimum of a set. Recall the supremum of a set, if it exists, is the least upper bound. So, if we subtract any amount from the supremum, we can no longer have an upper bound. The infimum of a set, if it exists, if the greatest lower bound. So if we add anything to the infimum, we no longer have a lower bound. The definitions we give proof of today capture this idea.

Thanks to Nasser Alhouti, Robert Rennie, Barbara Sharrock, and Lyndon for their generous support on Patreon!

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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[chinazaekperechinyere]

Thank you for this video 🥺
Made epsilon definition of infimum and supremum much more understandle

[hauntedmasc]

I'm honestly shocked you didn't say, "The proof for infimum is similar, " and smash-cut to the end card haha

[infinitescience]

Thank u so much, you made my concept clear, I was struggling in relating these 2 definition.

[digitalMathMechanic]

2:25. Very small correction, the if and only in individual definition is not the reason we need to prove in both directions, but rather because we want to show both definitions are equivalent. That is the definition on the left iff definition on right. I know that was a slip of tongue.

[catherineferris]

This video was so helpful! Struggling analysis student here lol

[divyamgaur]

After watching this, finally I am able to understand the complex definition of supremum and given in my book

[creatographer]

These are very timely. I have an exam this week :(, so thank you!

[nomachinesinthisroom]

Thank you so much for this content! I don't have to mention how clear and pleasantly it is presented (I did :|). I have a question. How does one come with the idea to subtract beta from both sides of the inequality? How do you think of this step? What is the reasoning? Thank you very very much!

[creatographer]

Do you have any videos on proving sets are compact? By the definition of compactness and also by the Heine-Borel theorem? I dont really understand that topic. Thank you!

[modaloperator]

What happens if we replace strict inequality in the epsilon definition, by ≤?

[Catalinawolff]

Hey ! Why do wie subtract from Alpha Not only ß but also (a-ß) ? Sry didnt understand that

[chanellioos]

hey wrath of math do you have more real analysis videos for subscribers that are Patrons in patreon ?