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.
Definition of Supremum and Infimum: • Definition of Supremum...
Proof that suprema and infima are unique: • Proof: Supremum and In...
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[@chinazaekperechinyere1252]

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

[@WrathofMath]

Haha I was tempted! But leaving things incomplete bothers me a bit, and for anyone who didn’t quite follow every step of the supremum proof, going over the infimum quickly gives them another round of explanation!

[@tapendrabc904]

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.

[@infinitescience3224]

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

[@WrathofMath]

Glad it helped!

[@ekaril5060]

Thank you so for these videos!

[@WrathofMath]

My pleasure! Thanks for watching, if you haven't already - check out my real analysis playlist! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@michelsfeir]

This was very helpful. Cheers!

[@WrathofMath]

Glad to hear it, thanks for watching!

[@cjjk9142]

This is perfect

[@mahmoudalbahar1641]

Many thanks for this good video.

[@WrathofMath]

So glad it helped, thanks for watching and let me know if you ever have any requests!

[@mahmoudalbahar1641]

@@WrathofMath My request is please When it is convenient to you continue with Real Analysis course.

[@creatographer1]

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

[@WrathofMath]

Glad to help! Thank you for watching!

[@creatographer1]

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!

[@WrathofMath]

Unfortunately I do not have such lessons! I'd make some for you but I have not spent much time studying topology. I've studied it a bit, but still have more to go before I can teach about compactness of sets!

[@catherineferris4379]

This video was so helpful! Struggling analysis student here lol

[@WrathofMath]

So glad it was helpful! The struggle is worth it for this wonderful area of mathematics. Good luck as you continue to study it, and let me know if you ever have any questions! If you aren't already looking at it, be sure to check out my Real Analysis playlist: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@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!

[@cjjk9142]

The idea is to create our epsilon which we need to be greater than zero, taking away beta does this

[@modaloperator]

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

[@divyamgaur7236]

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

[@WrathofMath]

Glad to hear it, thanks for watching! If you're looking for more real analysis, check out my playlist! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@chanellioos]

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

[@Catalinawolff]

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