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.
Epsilon Definition of Supremum and Infimum: • Epsilon Definition of ...
Proof that suprema and infima are unique: • Proof: Supremum and In...
Proof the maximum of a set is the supremum: • Proof: Maximum of a Se...
Absolute value inequality result we mention near the end of the lesson: • Proof: A Useful Absolu...
Supremum and Infimum Example Exercises:
• Proof: Supremum of {1/...
• Supremum of the Union ...
• Proof: Supremum of {1/...
• Proof: Minimum of a Se...
• Proof: Infimum of {1/n...
• Prove Infimums Exist w...
Thanks to Nasser Alhouti, Robert Rennie, Barbara Sharrock, and Lyndon for their generous support on Patreon!
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I hope you find this video helpful, and be sure to ask any questions down in the comments!
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[@WrathofMath]

Check out my Real Analysis playlist for more! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Let me know if you have any requests!

[@egbujuoemmanuel47]

Why can't the infimum of natural numbers be > 1

[@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

[@xyz-fq8lv]

Very nice and completely understandable at once about the concept.

[@rachelmadoo3671]

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

[@WrathofMath]

Thank you Kakashi-sensei! Be sure to keep your Sharingan on my Real Analysis playlist, many more lessons coming! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Or if you're in a more rhythmic mood, don't miss my math raps: kzbin.info/aero/PLztBpqftvzxW7a66b0dJPgknWsfbFQP-c

[@rachelmadoo3671]

@@WrathofMath ahhhhhhggggzjsn okay!

[@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.

[@WrathofMath]

So glad it helped, thanks for watching!

[@JimbobFaz]

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

[@WrathofMath]

Thanks so much! Still a long ways to go with this analysis playlist, but I've enjoyed making the highest quality videos I can for the subject! I'd like it to be the definitive playlist on the topic when I am done.

[@JimbobFaz]

The way you explain the concept and rational before just diving into the proofs is very helpful in getting the idea across, and this ultimately makes understanding the proofs and definitions a lot easier. Your proofs are also very good and rigorous. I have a 1st class undergraduate maths degree and I still find real analysis pretty difficult. Im trying to watch your videos ATM as a refresher and to try deepen understanding as im starting to forget this stuff after not studying it for a while and when picking up an old text book on metric spaces it all seemed rock solid again! I get rusty so fast when I stop studying maths. The hardest bit in real analysis that I remember not understanding very well and struggling the most with was about function spaces and uniform convergence of functions (not sure if this is what isknown as functional analysis though?)Also, the in abstract algebra I found in fields was very hard, rings not as much, and group theory was fine.

[@mahmoudalbahar1641]

Many thanks for this good video.

[@user-wr4yl7tx3w]

so well explained!

[@GerardFunClub]

Thank you very much!!!

[@ayaudh]

Very nicely explained.

[@valeriereid2337]

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

[@WrathofMath]

Glad to hear it - thanks for watching!

[@abcdefghijklnopqrstuvwxyz5259]

Your lectures are great ...love them

[@WrathofMath]

Thank you so much!

[@OmarAhmed-ic4fw]

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

[@ibraheemmoosa]

Thanks for this. Great explanation.

[@WrathofMath]

My pleasure, thanks for watching! Let me know if you have any questions, and if you're looking for more real analysis check out my playlist! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@aishwaryapotdar1348]

crystal, got it, thank you

[@lalhriatpuiahmar5057]

Thank you for your great explanation

[@WrathofMath]

My pleasure! Thanks a lot for watching, and if you're looking for more analysis - check out my playlist! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@Hana-gj7mi]

Great video, thanks!

[@WrathofMath]

My pleasure, thanks for watching and check out my real analysis playlist if you're looking for more! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@cheatyhotbeef2636]

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!

[@WrathofMath]

Thanks for watching and great question! I'll mention I don't love doing a bunch of lessons with no particular context, but the most common situation is that someone is learning something in a class - where they're getting more context - and just need some extra help understanding a particular concept, which is why I think its worthwhile to present topics with hyper-focus on the concept itself, without spending too much time on context which the majority of viewers who find this lesson over the coming years will already have from their classes. In this lesson though I touch on perhaps the biggest use of supremum and infimum right at the end of the lesson! By taking the completeness axiom, which guarantees us the existence of infima and suprema of bounded sets, we can "complete" the rational numbers - filling in their holes with all the irrational real numbers. And this completeness is tremendously nice to have. In future real analysis lessons we'll see more powerful results involving bounded sets, and suprema and infima, which will hopefully make their value more clear! And of course, it all builds up a rigorous foundation for calculus which is some of the most useful math ever devised!

[@anjaneyasharma322]

Cheaty Hotbeef Someboddyhas reached the top So whatever he says or observes has to be of very high value. In today's times it can be for getting doctorate. Simply put in the given set of values or bounded interval there will be two values Highest is Supremum and Lowest is Infimum. As Cheatybeef says this observation is very very ordinary. If instead of interval a function is taken It will have some value like Function is convergent or divergent real roots or imaginary etc for analysis.

[@maxpercer7119]

this is the one question you are not allowed to ask a math teacher (or any teacher) , WHAT IS THE POINT OF LEARNING this? answer: one day you will find out (or not)???

[@placeholder6811]

@@maxpercer7119 As a topology teacher for 20 years, I wholeheartedly disagree with you. You are speaking from arrogance. Students who are learning new material don't even know what they don't know so it is nice to have something to ground their learning direction. Walking around a subject vacuously is one of the easiest ways for students to get disinterested. For shame if you are an educator.

[@hnr683]

very talented work and very clear

[@WrathofMath]

Glad to hear it, thanks for watching and check out my analysis playlist for more! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@archibongbenjamine8606]

Thank you this vidoes really brought me out of confusion

[@WrathofMath]

So glad it helped! Thanks for watching, and if you're looking for more analysis check out my playlist! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@user-wd2nx4lt1g]

very interesting

[@catherinegutierrez2024]

Big help! Thanks!

[@WrathofMath]

My pleasure, thanks for watching!

[@axeanshuman]

Nice and simple explanation 😇❤

[@WrathofMath]

Thank you, glad it was clear! Let me know if you have any questions, and check out my playlist if you're looking for more analysis! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@abhiramisudhakaran137]

Thank you!!

[@WrathofMath]

No problem, thanks for watching! Check out my Real Analysis playlist if you’re looking for more on the subject, many more lessons to come! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@user-zn6xb5ho8f]

great . i learnt this easily from you❤

[@WrathofMath]

Glad it helped!

[@christoskettenis880]

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

[@mustafamalik4211]

Excellent explanation.

[@WrathofMath]

Thanks, Mustafa! Let me know if you have any questions, and check out my analysis playlist if you're looking for more: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@caotriinh3236]

great explanation!

[@WrathofMath]

Thank you! Check out my analysis playlist for more, and let me know if you have any questions! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@RSA_Shock]

Brilliant video

[@WrathofMath]

Thank you!

[@user-qb4pz6ru1c]

nice video

[@classicchessplayer7068]

I'm starting to like analysis now, thanks.

[@WrathofMath]

Glad to hear it! I like it a lot, it's a fascinating subject! If you haven't already, check out my real analysis playlist: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Let me know if you ever have any video requests!

[@classicchessplayer7068]

@@WrathofMath sounds good thanks :D

[@marcushendriksen8415]

Too good! Subscribed

[@WrathofMath]

Thanks, Marcus! Be sure to check out my Real Analysis playlist: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Planning to build it up a lot more as this year continues!

[@marcushendriksen8415]

@@WrathofMath I certainly will! Thank you for the link :)

[@tinotendakanyandura1139]

Can you help on the sup and inf of a closed set union natural numbers

[@williamsishaya1429]

Thanks for this .... pls I need more explanation on limit and continues

[@WrathofMath]

My pleasure, thanks for watching! I am working my way through creating lessons on analysis, so we will get to limits and continuity soon enough. Check out the playlist I am putting together if you haven't already: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Lots more to come!

[@tanishpanjwani3117]

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.

[@shayaanrk]

Good explanation

[@WrathofMath]

Thank you!

[@citizencj3389]

These concepts are VERY useful in understanding compact sets which can also help understand uniform continuity. Triangle inequality is op af in analytical math.

[@WrathofMath]

Definitely! And that's a funny way to put it - I agree, it's super OP, and you'd never expect it!

[@musratjahan4452]

Thank you sir

[@WrathofMath]

My pleasure, thanks for watching! If you're looking for more analysis lessons, check out my playlist! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@jonathanabraham2655]

Hello doctor, here 8:22 is the set {1/n: n€N} an open or closed set? I think it’s closed as taking x=1 then any delta>0 makes x+delta beyond the set. Is that assumption right or not? Thank you😊

[@shepherdchiti8098]

how do you show that the set S=a/b E(0,1) is unbounded

[@mkthakral]

Thank God you are here.

[@WrathofMath]

I do my best, thanks for watching and let me know if you have any questions!

[@mkthakral]

@@WrathofMath Thank you so much.

[@sumittete2804]

Sir ...what are the supremum and infimum of (0,1) intersection {m+n√2 : m,n€Z}?

[@sandeepchaudhary8689]

Wow great video ,greetings from india 🙏

[@WrathofMath]

Thank you, from the US! Be sure to check out my Real Analysis playlist if you're studying the subject, many more lessons to come! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@sandeepchaudhary8689]

@@WrathofMath yes sir ofcourse thank you so much 🙏I'm crying 😭in happyness🙏

[@faizahbegum5257]

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

[@tanvirkaisar7245]

Sorry for the dumb question, but for the last example, if we take Q as our field instead of R, then should not the sup and inf be +1 and -1 respectively?

[@roshanshihab8515]

Q means rational numbers not integers

[@lukaslukas305]

Wow 👏

[@WrathofMath]

Thanks for watching!

[@goedelite]

Aren't these usually called the GLB and the LUB ? I like to use lim sup and lim inf for sets, but maybe that is just my habit.

[@existentialrap521]

Your little dude saved me. I was thinking of b0 and b as being in the inside of the set S rather than outside. Once you moved the little yellow line i was like oh sht, this mfer spittin. Apologies for the crude comments. They help me retain sht way better than normal. That's why this is an anon account. Gang sht. I was raised in the hood tho. shootings at the corner trap house. they was cookin dope. get u a girl that'll cook w/ u. a ride or die. sup(S).

[@kingraj2692]

Teach us the maximum and minimum of a set( non empty set)

[@Mrtrendybombs]

Sir induced relation plz

[@AmjadKhan-xo2ux]

Is it necessary for a supermum to be an element of ordered field.

[@WrathofMath]

Yes indeed! We need to know what numbers are being considered. So if S is the set, the supremum need not be in S, but it must be in the field S belongs to (which field this is can depend on context, since sets can belong to multiple fields we must specify the one we are considering).

[@faizahbegum5257]

i don’t understand why 0 isn’t the infimum in the first example, is it because it isn’t a natural number?

[@citizencj3389]

Infimums are the greatest lower bound. 0 is bounded above 1 so it is not an infimum...it is instead a minimum value. Remember the set of natural numbers do not include rational numbers. And 1 is the greatest lower bound of the set of natural numbers.

[@souravde6116]

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

[@he_is_mark]

this set is not bounded above, therefore there's no supremum

[@emmanuelcampos82]

What is the sup and inf of this S=(-∞, 5)?

[@tanishpanjwani3117]

sup = 5, inf does not exist

[@itsonlylevi]

Damnit! Now I want some soup.

[@WrathofMath]

Same!

[@efeberkeksi7301]

Isn't zero a naturel number as well?

[@WrathofMath]

Thanks for watching! Some people consider it one, I generally don't and in my experience most people don't (though it's far from a great majority). Sometimes the "whole numbers" are considered to be the set of naturals with 0 included. And if you're looking for more real analysis, check out my playlist! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@kazikmajster5650]

Hey! The Naturals start with 0!

[@WrathofMath]

Not on my watch!

[@085_juhaifajahan8]

​@@WrathofMath0!=1

[@maxpercer7119]

supremum = tightest upper bound , least upper bound, least max, best max infimum = tightest lower bound, greatest lower bound , greatest min, best min

[@randomvideos3628]

The first slide of an explanation should never be an esoteric definition... but a good video anyways.

[@WrathofMath]

Thank you! I disagree for these sorts of videos, because people generally find them by searching for them specifically, and so have already had some form of introduction to the concept. I like to have the definition on screen so anyone who just wants a definition can see it immediately, but I begin this lesson with a simple explanation of what each definition means. For a normal in-person math lecture though, I agree!

[@kaos092]

How is 1 the greatest lower bound of the natural numbers? The logic used to make one the greatest lower bound would also make 2 the greatest lower bound.

[@WrathofMath]

1 is the greatest lower bound of the naturals because a) 1 is a lower bound of the naturals, there is no natural number less than 1 b) there is no lower bound of the naturals greater than 1, we know this because any number x greater than 1 couldn't be a lower bound of the naturals, since 1 (a natural) is less than x. For example, 2 is not the greatest lower bound of the naturals since it is not a lower bound at all, the natural number 1 is less than it.

[@kaos092]

@@WrathofMath Are you basically saying 0 isn't a natural number?

[@WrathofMath]

Yes. If you include 0 as a natural, then it would be the infimum.

[@kaos092]

@@WrathofMath Wow thank you!

[@egbujuoemmanuel47]

Why can't the infimum of natural numbers be > 1 @wrath of math

[@WrathofMath]

That's because the infimum is by definition the greatest lower bound, and seeing as 1 is in the set of natural numbers, anything greater than 1 cannot be a lower bound.

[@egbujuoemmanuel47]

@@WrathofMath thank you 🙏