Definition of Supremum and Infimum of a Set | Real Analysis

What are suprema and infima of a set? This is an important concept in real analysis, we'll be defining both terms today with supremum examples and infimum examples to help make it clear! In short, a supremum of a set is a least upper bound. An infimum is a greatest lower bound. It is easily proven that we can refer to "the supremum" and "the infimum" of a set because if they exist they are unique.

Supremum and Infimum Example Exercises:

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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[dotunderin]

This guy literally just explained in 10 minutes what it took my lecturer 2hours to explain with literally the same examples, the last one being a project
You just made my day man🔥

[oldcowbb]

i thought this would be a really bad video when you were just reading the definition at the beginning, but the explanations and the examples following are super helpful

[zafaris]

I'm studying logic for computer science at university and this really helped! The visual explanations and examples helped the concepts of supremum and infimum to click in my mind.

[rachelmadoo]

from the moment you said the channel name I knew this was gonna be good. Also these videos are a life saver

[JimbobFaz]

These videos are the best I have found so far on real analysis. Great work! 👍

[cheatyhotbeef]

Just wondering why do we use supremum and Infimum? What good are finding the upper/lower bounds of a set? I think I’d solidify these lessons more if I knew what they were used in, but the explanation was really clear and nice! Keep up the great work!

[xyz-fqlv]

Very nice and completely understandable at once about the concept.

[archibongbenjamine]

Thank you this vidoes really brought me out of confusion

[piyushkarki]

Since the set S is taken from an ordered field F, is it not redundant to say that the supremum b of the set has to be a part of the ordered field? Since it is already a part of the set, it is definitely a part of the ordered field. What I am trying to get at is, are there situations where one could confuse the supremum to be something that is not in the ordered field?

[christoskettenis]

So, just take any real number, either included in the subset or not, that fits the definition of lower or upper bounds, either the interval is open or closed. Just as long as it's not infinity

[valeriereid]

Thanks for this excellent explanation of the supremum and infimum. This most certainly helped.

[DonPlazo]

great . i learnt this easily from you❤

[classicchessplayer]

I'm starting to like analysis now, thanks.

[souravde]

1 ) For N = {1, 2, 3, ...} Why can't we say infinity is the supremum?

[tanishpanjwani]

Just a quick question at 6:25 do you mean 0 is greater than n instead of 0 is greater or equal than n for all n belongs to the natural number set.

[OmarAhmed-icfw]

May you make a video on the completeness axiom, pleases!

[lt]

2:20
So the upper bound of S doesnt have to be in the set of S?

[sandeepchaudhary]

Wow great video, greetings from india 🙏

[faizahbegum]

for the last example why can’t the infimum be -sqrt (-1) as it’s less than 2