Definition of the Limit of a Sequence | Real Analysis

[@WrathofMath]

Support the production of this course by joining Wrath of Math to access exclusive and early videos, original music, plus the real analysis lecture notes at the premium tier! kzbin.info/door/yEKvaxi8mt9FMc62MHcliwjoin Real Analysis course: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Real Analysis exercises: kzbin.info/aero/PLztBpqftvzxXAN05Gm3iNmpz9SkVfLNqC

[@nihilisticboi3520]

The moment you finished explaining the 1/100 example, I felt I reached enlightenment. Beautifully explained!

[@WrathofMath]

That's the best I can hope for! Thanks for watching, I am glad it helped!

[@SequinBrain]

if only textbook authors knew the value of using numbers in math books...

[@hamadkhan2339]

Sir, this is the best lecture of this topic on u tube. Thank u so so much for explaining everything. This was hard to me but u explained it in a very easy way. why so underrated? You are really good, I hope that soon you will have subs in millions🙏🙌

[@WrathofMath]

Thanks for watching, Hamad! I really appreciate your kind words! It's a very important topic so I'm very glad you found my explanation clear! The long road to a million awaits us!

[@pxlbits6442]

I learned more in this video than my entire semester in analysis class lmao...

[@WrathofMath]

Well that's good and bad haha - I'm at least glad my video was helpful! If you're looking for more analysis, check out my analysis playlist: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Lots more analysis coming! And if you're in the mood for some tunes, check out my new math song that came out today! kzbin.info/www/bejne/ZoDVoGVujdCtmMk

[@stephen3418]

These intro to analysis videos are incredibly valuable. You have clarified the ‘setting out’ stage of the proof that my textbook simply failed to describe. How to choose an N? Thanks to your teachings, I can understand analysis and apply its techniques! I had thought that perhaps I had truly arrived at a level of maths which I’d find incomprehensible (at least without some miraculous discovery), but your teachings elucidated the simple technique of how to algebraically decide what N ought to be. So thank you so much! This will certainly be the most valuable lesson in analysis of my life! Such is the nature of this incredible inflection point!

[@manarmagdy2329]

I cryied because I couldn't understand that and my exam after 4 days but now it's more than clear. Really thanks from my heart

[@swastikadeb8605]

I was really struggling to understand the definition of limit clearly. I looked up everywhere on the internet until I found this video, such nicely explained with that amazing 1/100 example. It's totally clear now! Thank you sir. Hope this reaches to everyone out there.

[@bigollameo]

Now I finally understand what the "n>N" condition means. For the longest time I've been puzzled as to just what this relation requirement was trying to communicate. Should have been apparent years ago, but better late than never. Thank you!

[@WrathofMath]

So glad it helped, thanks for watching!

[@toxicfreeze-brawlstars121]

bro i still dont understand this n>N can you explain

[@toxicfreeze-brawlstars121]

i mean what is the big n here

[@blazobanicevic]

@@toxicfreeze-brawlstars121 not sure if I got it myself but as for how I understand it, n is the index of an element of sequence (an) and N is a number so when you say n>N I think it is saying that every next element of a sequence after number N has to be between the limit and number N(so if I say 1/n and N=100 then every element after a100-after 100th element of a sequence-has to be between limit and 1/100 for every n>N) hopefully I helped but as I said I dont understand it completely however this is the way I currently understand it to be.

[@nikeshlilani5215]

this is by far the best lecture on this topic on youtube..thanks a lot sir !

[@WrathofMath]

That means a lot, thanks so much! I am glad it helped and let me know if you ever have any questions!

[@nateman44ify]

My professor just went over this topic and I just could not follow along or understand what he was trying to say. But this actually makes sense, because of you im actually completing my homework. Thank you!

[@WrathofMath]

So glad you found it helpful! Thanks for watching and let me know if you ever have any questions!

[@DoEvenly]

this video is the best video about limit I have seen in this year 😮

[@ConstanceShayamano-ei7fc]

Yu the best online teacher I have ever encountered with

[@AmeenaMuddasir]

Ahh great sir ! I was focusing on what is actually written in the book but got really upset because I was wasting my own time ! Then it clicked in my mind Ka I should watch a video and luckily I got you and your video ! Very good job sir thank you ❤

[@WrathofMath]

Thank you - glad it helped!

[@IdoYos]

Best explanation I have found so far!! If you could make videos explaining linear algebra as well it would be phenomenal!! Thank you so much!!

[@WrathofMath]

So glad to hear it, thanks for watching! You can find more analysis lessons in my Real Analysis playlist: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli As for linear algebra, I definitely plan on getting to it, but it will be some time. It was not one of my stronger subjects in college, so I have more than just brushing up to do!

[@IdoYos]

@@WrathofMathHaha tell me about it 😉 Thanks again for a great intuitive explanation!!

[@le_great_escape]

I finally understand where my professor for 1/epsilon. THANK YOU!

[@bactran7799]

amazing explanation of the limit of sequence definition that blows my mind. Thanks a lot Wrath. Please keep doing your great works

[@WrathofMath]

You're very welcome, thanks so much for watching! If you're looking for more analysis, check out my playlist! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@Anonymous-rv3gz]

Man you saved my 48hrs of stress...... May god bless you man.....

[@WrathofMath]

So glad it helped. Thank you very much for watching!

[@liss35]

perfectly explained, well defined that makes things clearer than on the mere printed definition from the book.. Thanks for this tutorial video it really helps a lot.. 👏👏👏

[@WrathofMath]

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

[@samaritan2899]

Math 101 in Uni and you make it sound so simple. I like the style

[@WrathofMath]

So glad to help - thanks for watching! I hope you'll continue to find my analysis playlist helpful, good luck!

[@aqsamaryam8580]

I think it is the best explanation for anyone to understand this concept.....

[@WrathofMath]

Thanks a lot, I'm very glad to hear that! 😊

[@hanskywalker1246]

Finally somebody explained why we have to use N instead of just always n, thanks!

[@WrathofMath]

Glad to help, thanks for watching and let me know if you ever have any questions!

[@hanskywalker1246]

​@@WrathofMathI have a question: 9:31 can't we also say that |an-L| is greater zero because like the limit will actually never reach the final value?

[@zb5775]

Your videos and your teaching skills are outstanding. Thank you for all your work.

[@WrathofMath]

Thank you!

[@BlaqueT]

This video was a good step to help my understanding, as this epsilon definition of a limit has confuded me so much in the beginning of my real analysis course. Thank you so much!

[@WrathofMath]

So glad it helped! Thanks for watching and check out my analysis playlist for more! Let me know if you have any questions! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@existentialrap521]

We makin it outta the trenches with this one. That visual helped a ton. Made the absolute value of an-L make sense.

[@snclassical8263]

I hated Real Analysis and THANK YOU! for giving me new hope!!

[@victoriacorcimaru1731]

Best, most intuitive explanation!!! A must watch )))

[@WrathofMath]

Thank you! Lots more analysis lessons to come - thanks for watching!

[@fatimaintekhab7357]

Thank you for explaining the definition with a graph! I was really struggling with this❤

[@marcusnzioka6624]

thanks for this explanation, I have struggled looking for good content like this

[@WrathofMath]

Thanks for watching, Marcus! If you're looking for more analysis, check out my Real Analysis playlist: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli And let me know if you ever have any video requests, lots more analysis lessons coming!

[@OGMarkuss]

Hello Wrath of Math. I'm preparing for a test righ now. The explanation in masterly presented, although I don't understand there part where you wrote N > 1/e . Could you, please, briefly explain why is that so?

[@VocalBeast]

Man thank you so much. Was pretty stumped understanding this but you explained it so clearly, thank you:)))

[@WrathofMath]

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

[@alfieplant6927]

Perfect explanation. I understand completely now

[@WrathofMath]

Glad to hear it! Thanks for watching, and check out my real analysis playlist if you're looking for more!

[@chad2336]

best explanation so far, great video man!

[@WrathofMath]

Thanks Giga-Chad!

[@arabmohcin-13]

wow ! is just you make eveything obvious ! love you bro

[@braverodger]

you explain very goodly. thank you!

[@WrathofMath]

Glad it was clear, thanks for watching!

[@pamodpeiris154]

Best explanation of limit..

[@WrathofMath]

Thank you, glad it was clear! And if you're looking for more analysis, check out my playlist! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@mathechne]

just the classical, very good, thanks

[@evgenirusev818]

Super clear and intuitive. Thanks!

[@WrathofMath]

So glad it was clear! Thanks for watching!

[@J078_]

Gud work sir....keep updating with sch wondrfl classes from real analysis....

[@WrathofMath]

Thank you! Will do, I have many more analysis lessons planned. Let me know if you have a specific request you'd like to see soon!

[@dark.violin]

You are really good at explaining. Thank you so much :)

[@michelleyuangu7462]

So well explained. Thank you so much

[@WrathofMath]

Thanks for watching!

[@aniketgupta1340]

at 10:40 , i guess there is a mistake cuz for some a>b and a>c, we cant establish the fact that b>c or c>b

[@MuhammadAhmed-hu6dk]

Holy why tf the book makes it so hard on us I mean it such a simple concept thanks you

[@Carol.26]

Thank you so much Sir! Your explanation was really helpful!

[@WrathofMath]

Glad it helped!

[@deisooja2961]

this was an excellent video about limit of sequences. I was on a lecture and didnt understand shit about it. Really thanks man !!!

[@WrathofMath]

Glad to help!

[@Teja-tm4zi]

excellent vid! thank you so much for the effort you put into these videos!

[@WrathofMath]

Thank you for watching!

[@ΔημήτρηςΖυγάς]

Thank you for making this clear! Could you maybe tell me how this example would work if we could guess the limit L of the sequence? How would we take out the absolute value, etc?

[@mahmoudalbahar1641]

Many thanks for this good video.

[@rishabhsharma1982]

Great video, incredibly helpful!

[@citizenofreality2751]

WOW! Thanks for all the work!

[@arthurbouchardet6932]

This is such a good video. Thank you, for real. Liked and Subscribed.

[@WrathofMath]

Appreciate it, thanks for watching!

[@danielcamacho5434]

amazing work dude! thank you so much!

[@WrathofMath]

Thank you, it's my pleasure! Be sure to check out my Real Analysis playlist if you haven't, many more analysis lesson to come! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@sukhofruktt]

you are a life saver!!! thank you so much

[@mohamedbenjbara8663]

thank you for the approach 😍😍

[@WrathofMath]

Glad to help, let me know if you have any questions!

[@successorof68]

I know the video is 3 years old but if you see this can you show an example where the definition for convergence of a sequence breaks. Maybe something like this: Prove 1/n limit is 1 when n goes to infinity. We know 1/n approaches 0 as n goes to infinity but would like to see how the proof breaks

[@doctoralejandro1334]

I'm wondering the same

[@hilmakara4907]

May you please do another video with examples involving accumulation points

[@mauryaneha589]

This vedio very helpful for me. Thank u so much sir

[@WrathofMath]

You're very welcome, thanks for watching!

[@mauryaneha589]

@@WrathofMath sir plz upload some examples of limit of a sequence

[@khalilmohammed2297]

Thank you so much, you are brilliant

[@karlathebarla]

Excellent explanation!!

[@WrathofMath]

Thanks Karla! If you're looking for more analysis, check out my playlist and let me know if you ever have any questions! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@zyshri]

Thankyou so much sir 🌸✨❤️

[@WrathofMath]

Glad to help - thank you for watching!

[@sahelaria993]

Thanks that was very helpful

[@WrathofMath]

Awesome, thanks for watching! Lots more analysis videos on the way, let me know if you ever have any questions!

[@aubreydebruyn84]

I consulted YT vids and RA books, did not get the N part. As so as you put on a graph, the lights turn on. Thank you so much.

[@WrathofMath]

Glad to help - thanks for watching!

[@AryanSingh-tt4kt]

could u please attach a word or pdf file for the format of the answer to be written like "Let E>0.................Hence, proved"

[@EduardoAdameSalles]

Thank you! BTW, what is this board software you use with the drawing tablet?

[@WrathofMath]

You're very welcome, thanks for watching! I use Notability on iPad Pro! It works great! If you're looking for more real analysis, check out my Real Analysis playlist and let me know if you have any video requests! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@EduardoAdameSalles]

@@WrathofMath Thanks for answering. I'm actually having calc 1, but my professor have decided to start by Real Analysis. In fact, calculus isn't that hard, but I'm not sure about my skills in RA. I'll certainly be watching your videos

[@IODell]

Great explanation. Thanks.

[@WrathofMath]

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

[@hanskywalker1246]

I have a question: 9:31 can't we also say that |an-L| is greater zero because like the limit will actually never reach the final value?

[@mohamedaliekoroma5040]

This is my question; Sum the following series “the sum of n where r = 1 ~ r(r-1)

[@mohamedaliekoroma5040]

I enjoyed the lesson and I learned a lot. You don’t mind me sir; I have a question that I want you to help me solve.

[@ioannaathineos5269]

Master ! Thank you

[@WrathofMath]

Glad to help!

[@MathematicsApplications56]

Thanks alot

[@kehalitadesse5165]

Thanks man that helped a lot

[@WrathofMath]

So glad to hear it! Thanks for watching and if you're looking for more analysis, check out my real analysis playlist: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Many more on the way!

[@sukrandinc5836]

Perfect, thank you

[@WrathofMath]

Glad it was helpful! Let me know if you have any questions and check out my analysis playlist for more! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@Godfather-ps6zv]

Man love from India

[@WrathofMath]

Love back from Cape Cod!

[@azizahsaswandila6708]

Thank you Sir

[@WrathofMath]

Glad to help! Thanks for watching and check out my analysis playlist if you're looking for more! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@athethird4945]

Aahhhhh thank you, When am Amat major attempts pmat

[@RamyChenafi]

what do you think about n=1+[1/e]

[@EvrYRAp]

holy. you are so good

[@user-xg8zi9tm7u]

You are best

[@WrathofMath]

Thank you!

[@cyrillamphosa3285]

Great stuff

[@Itsjustj82]

Why do we say that the bigger N is greater than 1/epsilon? (N>1/E)? please someone answer, i'm confused..

[@hatemelsaudy8892]

Me too 😢

[@0.O0x]

It is arbitrary guys...

[@WrathofMath]

What's arbitrary? Certainly not this definition! But the positive epsilon, yes!

[@uniquename2386]

Can someone explain to me why he wrote that big n is greater than 1/epsilon

[@navyatayi6956]

Here how is the value of epsilon considered? Is it a value related to the series or is it any number?

[@WrathofMath]

Thanks for watching and epsilon is an arbitrary positive number. For any positive epsilon, a convergent sequence will eventually get within epsilon of its limit. Does that help?

[@WiskundeTraining]

what is the difference between the N-epsilon definition and the epsilon-delta definition? why we need to use the N-epsilon definition for a sequence and not the epsilon-delta one?

[@WrathofMath]

Thanks for watching and good question! The N-epsilon definition is for limits of sequences as n goes to infinity. The epsilon-delta definition is for limits of functions at their accumulation points.

[@mo8046]

What could go wrong if I tried to prove that the sequence converges to any other number? What could go wrong if I tried to prove that, for example, (n+1)/n converges to 3 rather than 1?

[@WrathofMath]

Thanks for watching and that's a really great question! You should give it a go! Try proving (n+1)/n converges to 3 the same way you normally would. As you work with the expression | (n+1)/n - 3 |, you'll eventually see an expression clearly greater than or equal to 1 for all n, so you'll not be able to make it arbitrarily small. Then, you could say for epsilon = 1 and all n, we see what you just showed about | (n+1)/n - 3 | being at least 1, and so clearly there is no N that will make | (n+1)/n - 3 | less than epsilon for all n > N; proving the sequence does not converge to 3. I may make a video on this, it's a great thing to see when we begin working with the limit of a sequence definition!

[@mo8046]

@@WrathofMath Thanks for the great explanation!

[@noahniklaus5070]

Thank you

[@WrathofMath]

No problem, thanks for watching! Looking forward to making some more real analysis lessons.

[@sumeyye7014]

How can we assume that N is bigger than 1/Epsilon ?

[@WrathofMath]

N is a thing that we can choose, it effectively represents how far in the sequence we need to go in order to make the rest of the math work. When we do an epsilon proof, we're proving that no matter how small epsilon is, we CAN go far enough in the sequence so that all terms are within epsilon of the limit. N will generally depend on epsilon, because the smaller epsilon is, the further in the sequence we need to go in order to be sufficiently close to the limit.

[@guranahemanthkumar8976]

Thank you So much :)

[@WrathofMath]

My pleasure, thanks for watching! Be sure to check out the whole Real Analysis playlist if you haven't, many more new lessons to come! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@Groove_Lyrics]

Someone please explain why he changed < sign to n>1/epsilon

[@WrathofMath]

Not 100% sure what you're talking about (timestamp would be helpful) but if you're talking about going from (1/n) < epsilon, then we can invert both sides as long as we change the direction of inequality, thus getting n > (1/epsilon), telling us how big n needs to be for our argument to work.

[@syedalauddinbukhari9801]

help a lot man thankx

[@WrathofMath]

My pleasure, thanks for watching!

[@brainsofmathematics]

I can't explain how much u help me

[@navyatayi6956]

And why is epsilon any positive number close to 0? Why not a negative number?

[@WrathofMath]

Let An represent an arbitrary term from a sequence which converges to L. Epsilon is positive because it is used to describe how the distance between a convergent sequence and its limit gets arbitrarily small. The distance between An and L is |An - L| which is always at least 0 by definition. Thus, it is arbitrarily small positive numbers which the distance between An and L must get less than, since by definition it will always be greater than any negative number. For a convergent sequence, |An - L| will eventually be smaller than any given positive number epsilon. Does that help?

[@navyatayi6956]

@@WrathofMath yes it does. Thank you very much. I understand now. Amazing video btw :)

[@ekemm]

Thanku, and i subscribed 😁😁

[@WrathofMath]

My pleasure and thank you for watching and subscribing! Let me know if you ever have any requests!

[@rodrigodeboni4108]

YOU A G

[@bernatbosch]

2:16 "then, as like in this example, our sequence approaches zero, I can guarantee that..." and for me that is the problem of this type of definition of what a limit is: that just works if you already know in advance what the value of the limit is. Because otherwise, how could you calculate the absolute value of the difference between a number and L when you don't know yet what is the limit of the function as it approaches to...etc. A good definition of a limit L should be developed without having to know in advance the value of L. Otherwise, why the hell you need the definition if you are already sure that a certain number is indeed the limit you are searching for. Another idea that bothers me is that this particular way to prove seems just adapted for the case that we are wanting to know the limit of the sequence when it grows to infinity, but does not seem to be a good way to prove the limit of a sequence when it approaches a certain index n (lets say n=0 or n=1).

[@WrathofMath]

It should be unsurprising that to prove a limit converges to L, we need a candidate for L. There are various other ways this might take form, but regardless it is not like we have to KNOW what a limit is for sure before we can attempt this sort of proof, we need only a candidate, a number we think the limit is. With specific sequences, computation and/or intuition can produce a reliable candidate. In other situations of analysis that are less particular, a candidate may be rather abstract, and thus a proof of the convergence is very important. In our proofs of all the sequence limit laws, we don't KNOW the limit of a_n + b_n is a + b, but we suspect it is and so proceed with the proof to settle the matter. All that said, it is no doubt a significant and necessary weakness in this definition that you point out. The fact we must have some idea what the limit could be to use the definition is an obstacle. Later in my analysis playlist are several videos about Cauchy Sequences, in which we develop an equivalent definition of a convergent sequence that does not depend on the limit at all. Regarding the n to infinity point - there is no way, as we have defined sequences, to take a limit anywhere other than infinity. For any sequence a_n, if we consider the "limit as n approaches 4" for example, this doesn't really mean anything. The closest n gets to 4 is a_3 and a_5. In each case, either a_3 = a_4 and/or a_4 = a_5. Or, there is some distance between these terms. This is to say, a_4 either does not get approached by its neighboring terms, or it equals its neighboring terms. In a functional limit this is different because we could take a limit as x approaches 3 since x actually CAN approach 3. x = 2.9, 2.99, 2.999, etc. But for sequences, n is discrete, and can approach nothing other than infinity. Hope that's helpful!

[@noorjahanjaleel8116]

can you make vdios on properties of infinite series of real analysis

[@WrathofMath]

Thanks for watching and I have more real analysis lessons in the works. If you have a request for a specific property of infinite series I can try to hurry that one along!

[@realworld2716]

What is N?

[@tashfeenamahmud4391]

but what if a_n

[@WrathofMath]

Thanks for watching and good question! Yes, the definition works just the same. Remember we use the absolute value, we require that | a_n - L | is less than epsilon. This is the distance between a_n and L, and it does not matter whether a_n or L is negative. Their distance, given by | a_n - L |, will always be nonnegative.

[@si_monster7365]

10:39 Why does the big N have to be bigger than 1/epsilon?

[@WrathofMath]

Thanks for watching and for the question! In the scratch work we did, we saw that n being greater than 1/epsilon will give us the result we want. However, by convention/definition, we don't take n to be greater than whatever we please, we only have that n is greater than N. I think of N as some point in the sequence, after which we get what we want. So since we want n to be greater than 1/epsilon, and we will have n > N, and we can pick what N is greater than, we need N to be greater than 1/epsilon so that n is as well. Then we're saying as long as we pick N to be greater than 1/epsilon, all terms of the sequence after the Nth term will satisfy our desired inequality. So for all n > N, |1/n - 0| < epsilon. Does that help?

[@si_monster7365]

@@WrathofMath Very clear, thanks!

[@muhtedbaig7011]

@@WrathofMath thanks for the explanation I really appreciate it. I just had one query. Since n>N, we can also consider N to be *greater than or equal to* 1/epsilon right? Adding that *or equal to* shouldnt affect it as n>N. I would be glad ify help me clear this confusion :)

[@bernatbosch]

From 10:32 to 11:52 the reasoning is unnecessary because at 10:32 it was demonstrated that any n greater than 1/epsilon would match to the inequality.

[@WrathofMath]

True, but the introduction of a definition like this is not the time for proceeding quickly in an explanation. I am deliberately redundant at times for important definitions.

[@SerielxSylv]

can N equals epsilon?

[@sayakray4740]

epsilon is a very small number close to zero. Whereas in the definition you need a number n such that n>N and the distance between f(n) and the limit should be less than epsilon. Now for us to get this value of limit we must increase this value N as much as possible so that we can go more closer to the value of the actual limit of the sequence. That is why N can't be equal to epsilon. You can always put them to be equal but then you won't be reaching any close to the limit.

[@SerielxSylv]

@@sayakray4740 thanks for replying, im confused because i found some youtbe videos that put them = instead of > ... but u cleared my doubts thanks a lot👍