 That is interesting. Let us probably say that also. Now, a n is a sequence which is given to you a 1, a 2, a 3 and so on, a n and so on. Here is something which happens very often. You are standing in a queue and suddenly somebody with some tag or something comes. A special queue is made for you. You are VIPs. A separate queue. So, what you are doing? From that ordered collection, you are picking out some members. But, he is not allowed that after he has picked to the 10th person, he can go back and look at a army which is left and he will not allow you. Once you have taken it out, go ahead and pick up whichever you want. So, you are allowed to pick up elements of a sequence in an increasing order, whichever you like. For example, you can pick up this one. Now, go on. You can pick up this one. Go on. You can pick up this one. So, this one, you can call it as n1, your selection. Keep the hierarchy in the queue. Do not come back. Pick up the second one. This is an second selection, third selection, an3 and so on. So, a general ank, you can form a sequence by picking up elements from the given sequence. But, keeping in mind n1 should be strictly less than n2. n2 should be strictly less than n3 and go on. So, such a thing is called a sub-sequence. So, this is called a sub-sequence where n1 is less than n2 less than n so on. So, what I am doing? So, pick up a strictly monotonically increasing sequence of numbers n1, n2, n3 and so on. According to that, pick up the elements of the given sequence. That gives you a sub-sequence. Now, if a sequence is convergent, what can you say about the sub-sequence? Obviously, it should be because if those a n's are coming closer to a value l, part of that also should come inside. So, first fact, try to write a proof yourself, if you like, that if a sequence a n is convergent, that implies every sub-sequence must also converge to the same limit. Can I say converse is true? Of course, we will look at the sequence minus 1 to the power n. I pick up the odd terms, they converge to plus 1, even terms converge to minus 1. But, supposing I put a condition that every sub-sequence has to converge to the same limit, then it is obvious it should happen that your sequence itself should converge to that same value. So, this is a theorem. Again, we will not prove it, we will not ask you in the exam, but it is a simple obvious fact intuitively quite clear. A sequence converges to a limit l, if and only if every sub-sequence converges to the same limit, namely l. So, with that intuitive understanding, let me go over and something. Here is something. So, let us assume a n is a sequence. So, we just now observed a n convergent, if and only if every sub-sequence converges to the same value. It is quite interesting. Try to think of a proof yourself. You will see how logic is playing a role there. See, here is something which all of you should understand. If you want to understand when a statement is true, you should understand when the statement is false. This is the effect of mathematics and of life. If you want to know what is true, it is as good as knowing what is false. If you do not know what is false, you will not know what is true. Think about it. The logic is played in mathematics also. If you want to say some statement, see, I think let me elaborate that before writing it somewhere here. Where do I write it? Let me write it here. a n converges to l. That is a statement. That is same as saying for there exists some l. I think we did that last time. l belonging to r such that a n minus l is less than epsilon for every n bigger than n naught. That is convergent. So, the truth of the statement that a n is convergent is equivalent to writing this, that there is a number l, which is going to be the limit. What does that mean? For every epsilon, there should be a stage after which a n should come closer to. If this statement is not true, what does it mean? So, there is a now the other way around. If you want to understand what is true, you should understand what is false. So, what is the falsehood of this statement? a n is not convergent. That means what? The first thing was there exists some number l. So, this should go bad. That means what? That means for every l, you may have a problem later on using this slide because something is coming somewhere. So, let me I think let me not do that. Let me go over to here. So, a n converges to l is equivalent to saying there exists l belonging to r such that r and there exists some n not belonging to n such that mod of a n minus l is less than epsilon for every n bigger than n not. And I want to say that if this statement is not true, a n is not convergent. So, what is that equivalent to? So, first of all this statement should go bad. So, this is a combination of many statements. First one is there is some l such that something happens. So, it should go bad. That means if it is not going to be true for one particular l, that means for every l, it should be bad. So, for every l belonging to r, what should happen? Here it says there exists a stage. I should not be able to find a stage. That means for every n, whatever I think n is the stage, there exists some stage after that. So, let us call it as n 1 bigger than n such that this goes bad. That means mod of a n 1 minus l is bigger than or equal to epsilon. Whatever there is no number, that means for every number and what happened to that epsilon, whatever what happened to the epsilon, I did not write that. Such that for every epsilon was there, because epsilon did not come into picture. For every l given an epsilon, that means what? There exists at least one epsilon. So, there exists, I should write there exists some epsilon such that this goes bad. At least there is one interval so that you say that after that say everything is inside. No. At least one of them will go out and that is what we are saying. So this is what I am saying, trying to understand the negation of a given statement. a n is convergent. So try to do it every time whenever you find a theorem or something anywhere in your understanding. Even in your life, how do you compare something is true? You have to compare it with something is false. In probability theory, you will all be doing probability theory. What is the chance of this fan falling down just now or in a particular date at a particular time? Chance is probability. That is equivalent to knowing the probability of this not falling down. Both are equivalent statements. One is the truth that it will fall other is the falseness. It will not fall. So mathematically also you will find out these things coming back to you, because you are all ASI students. That is same. So let me not spend much time on it and go back to what I was saying is corollary we have proved. So I said every subsequence converges for a given sequence. I think the more things let me write here now. So a n is a sequence. So one we have already said, every subsequence converges if and only if the sequence converges. Now suppose a n is convergent. Why limit is not going to be important? So a n is convergent. Again, convergence means elements are going to come closer to it. Let us not bother about to what it comes closer. Let us see the effect of this. If L is the limit, it should come inside this. I can make it smaller is for every epsilon. Still a tail should come inside. So what should be happening to the terms of the sequence itself? Even if I do not know L the limit, but I know it is convergent. That means the terms of the given sequence must come closer to each other. As you progress, as n becomes larger and larger, a n should become closer and closer to each other. Because if they remain away, they are not going to converge. So it looks like an intrinsic property of the sequence convergence. A sequence convergence should imply that the terms are going to come closer to each other. Because if I look at the sequence, whether I know the limit or not, does not matter. I can just say that the terms are coming closer to each other. I look at that person and say it is honest. So that is the intrinsic property without verifying whether he has a communal case against him or not. That is the extrinsic property. Limit is something outside which is not given to you. But saying terms are coming closer is the property of the sequence I can verify. Let us give it a name. Let us call a sequence. We say a sequence a n is Cauchy. Cauchy was a mathematician and his name, I think you will find, he will come in your course also somewhere or the else. In mathematics, he comes left and right. In mathematics courses and statistics also he will come somewhere. We say it is a Cauchy sequence. If I can find there exists some stage n naught, everything is a property of the tail in sequences n naught such that a n minus a m, the distance such that if I should write that Cauchy, if every epsilon is bigger than 0, the distance is epsilon for every n bigger than. How close? As close as you want it, but for a tail. Given epsilon, you want this close, that will be the stage. If much closer you want, some other stage will be there. But this is the property of saying it is Cauchy and I said it is an intrinsic property. So, here is a claim convergent, convergent implies Cauchy. Cauchy as a human being, convergent is a property. It looks very odd writing convergent implies Cauchy means every sequence which is, now we are using them as adjectives. If a sequence is Cauchy, convergent, then it is also Cauchy. So, we are not using it as a noun, we are using it as a adjective. So, let us see how is it true? Now, here is convergent means there is L, there is L minus epsilon and there is, that is the neighborhood, that is the interval, say that for every n bigger than n naught, a n's are here. a n is here, a m is here, when n and m are bigger than n naught. What that obviously says, the distance between a n and a m is small. So, let us write it mathematically. How do we write it mathematically? If you like, we can write it mathematically as a n converges to L implies there exists some stage n 1 such that or n 0 such that mod a n minus L is less than epsilon for every n bigger than n naught. What I want a n minus a m, that is my target, but I know something about a n minus L and a m minus L. So, let us bring in add and subtract. So, less than or equal to over every n minus bigger than, this is true for all n and m. So, in particular for n and m bigger than or equal to n naught and that says is less than 2 epsilon, because each is less than epsilon because of convergence. Is it okay? Because of this property, each is less than epsilon. Now, this is a only, what I call it cosmetic surgery. I do not want 2 epsilon, I want only epsilon to look it nicely. So, I will go back and say it epsilon by 2. So, it will be epsilon by 2 by 2 that is epsilon. That is a minor thing because epsilon is arbitrary. So, I can change it to anything I like to start with. So, that says every convergent sequence is Cauchy. So, Cauchy-ness is a necessary condition for a sequence to be convergent like boundedness. Question asked now, can I say if a sequence is, if the terms are coming closer and closer, will the sequence converge? For rationales, it is not true. For reals, it is true. So, we want to prove with theorem that every Cauchy sequence is also convergent. So, it is an equivalent way of saying convergence. Sometimes you are not interested in knowing the limit of the sequence. You are only interested in knowing the property of the sequence. Their Cauchy-ness is very useful because you do not have to bother about the limit. You have to only bother about given the sequence intrinsically look at whether the terms are coming closer or not. So, we want to prove a statement that every Cauchy sequence is also convergent. So, we will prove that probably next lecture because there are only three minutes and we cannot prove it. But here is something I want you to sort of have a look at it. Given a sequence a n, there is one way of visualizing it. I want to visualize this sequence, not as points on the line, but as if their heights of poles a 1 is some height, a 2 is some height. So, a 1, a 2, a 3, a 4, may be a 5 and such things. Imagine the sequence to be saying the height of something, a building probably or something. So, when will you say a sequence is increasing? Imagine they are buildings for example. You are able to see the top of this building. Increasingly, you are able to see the top of the next building also. This building does not obstruct the view of the next one. Everything is visible and you want to say it is monotonically decreasing, then everything below is visible. If it is something like this, you cannot say that this is increasing because with something is obstructed in between. But at least it looks possible. Let us start with something and let us see the next building which is visible to you and the next building which is visible to you and then the next. If I pick up these buildings, what I will get? I will get a subsequence which is monotonically increasing or sometimes I can see this and then probably I forget what this building then I can see. If this building is not there, if this is not there, then I can see the next one here which is smaller than this. Probably there is something smaller than this and then if I do not forget about these ones, then I can see that also. Then I will be picking up something, a subsequence which is monotonically decreasing. It seems to be a fact that every sequence has either a monotonically increasing subsequence or a monotonically decreasing subsequence. This picture seems to suggest to me that. We will prove that next time more mathematically that every sequence has got a subsequence which is either monotonically increasing or monotonically decreasing. Now, for a convergence, I need bounded. Suppose I am given a bounded sequence, then combine it with this result. Now, there is a subsequence which is monotonically increasing or decreasing and the original sequence is bounded. Then this must be bounded. So, every sequence which is bounded will have a convergent subsequence. That is again an important theorem. It is called Bolzano-Wittes' property. Now look at a Cauchy sequence. To say that a Cauchy sequence is convergent, it is enough to prove a subsequence is convergent. The terms are coming closer. If a subsequence is converging, the sequence itself must be coming closer to that value because they are also coming closer to each other. So, another fact that a Cauchy sequence is convergent. To prove this, it is enough to produce a subsequence which is convergent. Now, a Cauchy sequence has to be bounded also because they are coming closer and closer. Given epsilon only finitely, many can be outside. Again, it does not matter where. Again, a Cauchy sequence is going to be bounded. Cauchy sequence is bounded. Cauchy sequence has got a monotonically increasing subsequence. By the Bolzano-Wittes' property, it must converge. So, a Cauchy sequence has got a convergent subsequence. Hence, the sequence itself must converge. That will prove Cauchy-ness is equivalent to convergence. So, I have already given you a trailer of the next class. So, we will do it next time. Thank you.