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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!
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: ru-vid.com/group/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! ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-0Oro28Xkzbg.html
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!
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.
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!
@@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.
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?
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!
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 ❤
So glad to hear it, thanks for watching! You can find more analysis lessons in my Real Analysis playlist: ru-vid.com/group/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!
You're very welcome, thanks so much for watching! If you're looking for more analysis, check out my playlist! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
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! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
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! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
Thanks for watching, Marcus! If you're looking for more analysis, check out my Real Analysis playlist: ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli And let me know if you ever have any video requests, lots more analysis lessons coming!
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! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
@@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
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?
Thanks Karla! If you're looking for more analysis, check out my playlist and let me know if you ever have any questions! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
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! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
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
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?
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.
My pleasure, thanks a lot for watching! If you're looking for more real analysis, check out my playlist! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
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!
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!
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.
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.
So glad to hear it! Thanks for watching and if you're looking for more analysis, check out my real analysis playlist: ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Many more on the way!
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.
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?
Glad it was helpful! Let me know if you have any questions and check out my analysis playlist for more! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
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.
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! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
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👍
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!
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 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 :)
