Definition of the Limit of a Sequence | Real Analysis
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- čas přidán 16. 07. 2024
- What are convergent sequences, and what is the definition of the limit of a sequence? We introduce the definitions, with examples and a proof in today's video lesson!
In the definition of the limit of a sequence, we seek to capture what it means for a sequence to get arbitrarily close, or converge, to some limiting value. We do this by using an arbitrarily small "tolerance" value called epsilon. If you're in an analysis class, get ready to write "Let epsilon be greater than 0" an awful lot.
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I learned more in this video than my entire semester in analysis class lmao...
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: czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Lots more analysis coming! And if you're in the mood for some tunes, check out my new math song that came out today! czcams.com/video/0Oro28Xkzbg/video.html
The moment you finished explaining the 1/100 example, I felt I reached enlightenment. Beautifully explained!
That's the best I can hope for! Thanks for watching, I am glad it helped!
if only textbook authors knew the value of using numbers in math books...
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🙏🙌
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!
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.
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!
this is by far the best lecture on this topic on youtube..thanks a lot sir !
That means a lot, thanks so much! I am glad it helped and let me know if you ever have any questions!
amazing explanation of the limit of sequence definition that blows my mind. Thanks a lot Wrath. Please keep doing your great works
You're very welcome, thanks so much for watching! If you're looking for more analysis, check out my playlist! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
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!
So glad it helped! Thanks for watching and check out my analysis playlist for more! Let me know if you have any questions! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Thank you for explaining the definition with a graph! I was really struggling with this❤
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
Your videos and your teaching skills are outstanding. Thank you for all your work.
Thank you!
thanks for this explanation, I have struggled looking for good content like this
Thanks for watching, Marcus! If you're looking for more analysis, check out my Real Analysis playlist: czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
And let me know if you ever have any video requests, lots more analysis lessons coming!
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.. 👏👏👏
So glad to hear that, thanks for watching! If you're looking for more analysis, check out my playlist! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Great video, incredibly helpful!
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!!
So glad to hear it, thanks for watching! You can find more analysis lessons in my Real Analysis playlist: czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
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!
@@WrathofMathHaha tell me about it 😉
Thanks again for a great intuitive explanation!!
You are really good at explaining. Thank you so much :)
Super clear and intuitive. Thanks!
So glad it was clear! Thanks for watching!
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!
So glad you found it helpful! Thanks for watching and let me know if you ever have any questions!
Best, most intuitive explanation!!! A must watch )))
Thank you! Lots more analysis lessons to come - thanks for watching!
WOW! Thanks for all the work!
Man thank you so much. Was pretty stumped understanding this but you explained it so clearly, thank you:)))
Glad to help! Check out my analysis playlist for more and let me know if you have any questions! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
you explain very goodly. thank you!
Glad it was clear, thanks for watching!
best explanation so far, great video man!
Thanks Giga-Chad!
Yu the best online teacher I have ever encountered with
best video i have found for this so far
Glad to hear it! Thanks for watching and if you're studying real analysis be sure to check out my analysis playlist, many more lessons to come! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
excellent vid! thank you so much for the effort you put into these videos!
Thank you for watching!
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?
I think it is the best explanation for anyone to understand this concept.....
Thanks a lot, I'm very glad to hear that! 😊
Many thanks for this good video.
So well explained. Thank you so much
Thanks for watching!
Thank you so much Sir! Your explanation was really helpful!
Glad it helped!
Perfect explanation. I understand completely now
Glad to hear it! Thanks for watching, and check out my real analysis playlist if you're looking for more!
Excellent explanation!!
Thanks Karla! If you're looking for more analysis, check out my playlist and let me know if you ever have any questions! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
just the classical, very good, thanks
Gud work sir....keep updating with sch wondrfl classes from real analysis....
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!
amazing work dude! thank you so much!
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! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
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!
So glad it helped, thanks for watching!
bro i still dont understand this n>N can you explain
i mean what is the big n here
@@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.
thank you for the approach 😍😍
Glad to help, let me know if you have any questions!
Great explanation. Thanks.
My pleasure, thanks a lot for watching! If you're looking for more real analysis, check out my playlist! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Thank you so much, you are brilliant
you are a life saver!!! thank you so much
We makin it outta the trenches with this one.
That visual helped a ton. Made the absolute value of an-L make sense.
Thanks that was very helpful
Awesome, thanks for watching! Lots more analysis videos on the way, let me know if you ever have any questions!
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?
Great stuff
I hated Real Analysis and THANK YOU! for giving me new hope!!
Thanks man that helped a lot
So glad to hear it! Thanks for watching and if you're looking for more analysis, check out my real analysis playlist: czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Many more on the way!
Man you saved my 48hrs of stress......
May god bless you man.....
So glad it helped. Thank you very much for watching!
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
Best explanation of limit..
Thank you, glad it was clear! And if you're looking for more analysis, check out my playlist! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Master ! Thank you
Glad to help!
Thankyou so much sir 🌸✨❤️
Glad to help - thank you for watching!
May you please do another video with examples involving accumulation points
Perfect, thank you
Glad it was helpful! Let me know if you have any questions and check out my analysis playlist for more! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Thank you Sir
Glad to help! Thanks for watching and check out my analysis playlist if you're looking for more! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
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.
holy. you are so good
This vedio very helpful for me. Thank u so much sir
You're very welcome, thanks for watching!
@@WrathofMath sir plz upload some examples of limit of a sequence
help a lot man thankx
My pleasure, thanks for watching!
Thank you
So much :)
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! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Thank you! BTW, what is this board software you use with the drawing tablet?
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! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
@@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
Finally somebody explained why we have to use N instead of just always n, thanks!
Glad to help, thanks for watching and let me know if you ever have any questions!
@@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?
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
this was an excellent video about limit of sequences. I was on a lecture and didnt understand shit about it. Really thanks man !!!
Glad to help!
Thank you
No problem, thanks for watching! Looking forward to making some more real analysis lessons.
thank you
You're welcome!
This is my question; Sum the following series “the sum of n where r = 1 ~ r(r-1)
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?
Thanku, and i subscribed 😁😁
My pleasure and thank you for watching and subscribing! Let me know if you ever have any requests!
what do you think about n=1+[1/e]
You are best
Thank you!
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.
Glad to help - thanks for watching!
Aahhhhh thank you, When am Amat major attempts pmat
can you make vdios on properties of infinite series of real analysis
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!
I can't explain how much u help me
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?
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.
What is N?
Here how is the value of epsilon considered? Is it a value related to the series or is it any number?
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?
Man love from India
Love back from Cape Cod!
Why do we say that the bigger N is greater than 1/epsilon? (N>1/E)? please someone answer, i'm confused..
Me too 😢
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?
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!
@@WrathofMath Thanks for the great explanation!
YOU A G
❤❤❤❤
🔥
Thanks for watching!
Nyc sir
Thank you!
How can we assume that N is bigger than 1/Epsilon ?
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.
but what if a_n
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.
wonderful explanation..its very useful to me as a beginner teacher in real analysis. please give me instructions how to send donations in paypal im from the philippines
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).
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!
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.
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.
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10:39 Why does the big N have to be bigger than 1/epsilon?
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?
@@WrathofMath Very clear, thanks!
@@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 :)
can N equals epsilon?
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.
@@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👍
And why is epsilon any positive number close to 0? Why not a negative number?
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?
@@WrathofMath yes it does. Thank you very much. I understand now. Amazing video btw :)
TOOODO FALSO!!!
First! 😎
Thank you for watching! 😎
Too bad you couldn't provide a counterexample...
It is arbitrary guys...
What's arbitrary? Certainly not this definition! But the positive epsilon, yes!
you sound exactly like iDubbbz
I've received several voice comparisons, but haven't seen that one! I can't say I hear the resemblance, but if you do some others probably do too. I remember sitting down in my old college apartment after class to watch the Content Cop on Ricegum, those were the days!
@@WrathofMath yep the good old days