 So now this is something very intuitive, so let us just observe what is happening in this. A pendulum is swinging and what do we see when we see the pendulum. You are able to see the pendulum, swing. Now let us view this pendulum through a window, now I have put a window, you are able to see that. Window is big enough and you are able to see it. Let us make the window a bit smaller, what will happen, possibly if I make the window smaller I may not be able to see the full swing of the pendulum but what will happen if I wait long enough. If I wait because of gravity physics says that the swing of the pendulum will become smaller, even if my window is smaller I will be able to see the full swing. Let us make the window much smaller again, what will happen, maybe still I cannot see all the swing. But again I wait more longer enough, again I will be able to see. So what I am saying is I will be able to see the position of the pendulum in every window however small the window is. You will say the limiting position of the pendulum is the horizontal one, is that okay and that is what is convergence of a sequence. Position of the pendulum call it as an. So given a window of some length all the an's are inside, that is convergence of a sequence. I will write that as mathematically saying as follows, a sequence an is said to converge if there is a number l which is a real number such that given any window across length of the window. So given any epsilon look at that window, the length of the window is l minus epsilon to l plus epsilon. So what should happen, the pendulum should be visible maybe after 10 minutes. So there is a stage, so call that stage as an not such that an comes inside l minus epsilon and l plus epsilon and stays inside. So for all n bigger than n not, some stage or some weight everything is visible to me. All the terms of the sequence are visible to me. So mathematically saying that this happens right or you can write it as if you like, you can write it in terms of, there is a notion of absolute value of numbers, does everybody know absolute value of numbers for any number x real number, absolute value of x is defined as x if x is bigger than 0 and it is defined as minus x if it is less than 0. So this is essentially geometrically saying it is the distance of a point on the number line from the point 0, distance is always bigger than or equal to 0. So this is the convergence of a sequence of real numbers. So it is quite clear to everybody what is the convergence of a sequence, an is convergent if it comes closer to a value, so I have to say what value. So there exists a number l such that an comes closer to l, what does closer mean? I give you any window, so length of the window is epsilon right, so given epsilon bigger than 0, look at l minus epsilon n, l plus epsilon everything after some stage should fall inside it. So an is inside the window that means absolute value of an is distance from l, either this side or that does not matter, so absolute value of that an minus l distance is at the most epsilon from n bigger than n0 right. Another way of understanding this would be, you see an keep in mind that area thing, say an is an approximation to l right and what does an minus l indicate? It is the error you are making, actual value is l, you are taking an approximation a l, so the error is an minus l, that error I want small, how small? I will specify how small, so epsilon is a Greek letter for e okay, that is why one it has stuck as epsilon, so given the margin for error that is a distance for error epsilon, I will specify it should come that close, so an should come close to l, how close? That error should be less than this for all n bigger than n0 after some stage, so keep in mind for a sequence convergence depends only on the tail of the sequence, from some stage onwards for large n, it does not depend on the first few terms, there could be anything because what we are interested in is what happens for n very very large, after some stage if they all come inside that window for every epsilon, every window then good enough, what first 1 million, 1 trillion, it does not matter how many terms you have to forget, forget them right, so it is a tail of the sequence which is important for analyzing whether it is convergent or not clear, so that is what is called convergence okay, so this is what, so let us look at some examples I think, so let us look at the sequence 1 over n, so limit, so we write as limit n going to infinity that a n limit n going to infinity equal to l, so what is the meaning of that, that is only a symbol, that is only a rotational way of writing saying for every epsilon bigger than 0, there exist some n0 such that a n minus l is less than epsilon for every n bigger than n0 right, so let us look at the sequence 1 over n, so whenever you want to analyze whether a sequence is convergent or not you have some way of making a guess of that thing, what you think is possible happening, so let us most of the time you will have to do that and then prove it okay, so make a guess what happens to 1 over n, as n is becoming larger and larger, so n becoming larger meaning 1 over n 1, 1 by 2, 1 by 3, 1 by 4 becoming smaller and smaller, so guess, so this is my guess limit 1 over n is equal to 0 right, so how do I prove it, I have to prove it according to this, so let epsilon greater than 0 be given, then we want a n minus l less than epsilon for every n bigger than n0 right, so what is a n minus l here, we analyze that, the error we want to make it small, so let us analyze what is the error actually in our case, so that is 1 over n, so l is 0, a n is 1 over n, so this is 1 over n, so we want this to be less than, so we want 1 over n less than epsilon, what is epsilon, epsilon fix right, that is given to be a margin, that is same as saying that 1 over epsilon is less than n right, so that means given epsilon, I want n, so is that 1 over n should become less than, is it possible yes, because of Archimedean property right, a real number, given any thing I can go across by choosing n large enough, so that is Archimedean property says yes it is possible, so for that particular epsilon, for given epsilon become, there is a n0, so is that 1 over epsilon is less than n0 right, by Archimedean property, so that is same as saying that 1 over n0 is less than epsilon right, so if n is bigger than n0, then what happens to 1 over n, that is less than 1 over n0 and that is less than epsilon, so we got an a n less than epsilon right, that is how you write the proof that 1 over n has limit which is equal to 0. Let us do one more example, let us look at that sequence in the example that was minus 1 to the power n, n bigger than or equal to 1, so I have to make a guess, guess this convergent or not right, so what are the terms, so this is one way of right, look at the few terms of the sequence at least, it may give you a hint right, so terms are minus 1, 1, minus 1, 1 and so on, so if I look at the number line, here is 0, here is minus 1, here is 0, so the sequence is visiting minus 1 and 1 at every time point n and these are the only positions where it goes right, so intuitively it looks quite clear that it cannot converge to anything other than minus 1 or plus 1 right, if at all it converges the possibilities are either minus 1 or plus 1, so these are the only places occupied by it, so as a first step we want to show that, so guess is it does not converge right, so how do I write a proof of that, so we want to write a proof, so guess the sequence is not convergent, that is equivalent to instead of writing not convergent, not convergent, let us give it a name, we say it is divergent, something which is not convergent will be called as a divergent sequence, so we want to say it is a divergent sequence, so if possible let this sequence minus 1 to the power n divided by n bigger than or equal to 1 converge to let us say plus 1 or okay, we can do all of them 1 by 1, so here is 0, here is plus 1 right, here is, now intuitively if the sequence has to converge it has to come closer to some value and we are saying it is coming closer to the value plus 1 after some stage, but I know that whatever stage I choose right, after that positively it is going to come to minus 1, it is going to come away from it right, so that says to me that let us take epsilon equal to, epsilon bigger than 1 such that 1 minus epsilon is bigger than 0, so let us take here is 1 minus epsilon, I want to show it does not converge to plus 1, I have to say that there is a, okay, so this is important, saying something is not convergent what does it mean, so that is, let me discuss that first, if this thing does not happen, my guess is the sequence does not, then what I have to do to prove convergence given epsilon I have to find a stage n not, to prove it is not convergent what I have to do, I have to show that whatever l I choose right, whatever l I choose this thing should not happen, that means what, that means I should be able to, there is, if I want no l to be the limit right, that means I should be able to find at least one epsilon, one window right, so that whatever stage I say something goes out of it, everything after that is not visible, so something goes out of it, so let us write that that is important, so before I write everything a n does not converge to l, a n no I think better way of writing is a n is not convergent is equivalent to saying for every l bigger than 0, for every l which is a real number, see limit exists means there exists one l, I do not want that to happen that means for every l something should go wrong, what should go wrong, for every epsilon there exists some epsilon bigger than 0 such that for every stage n not, there exists some n bigger than n not with a n not, not belonging to l minus epsilon to, is that okay, saying that convergence means there is a l for which something is true, false for every l something is false, what is false for every l I can find at least one window that means there exists some epsilon, so that if I look at that window and look at any stage n not then there is one stage at least after that which goes out of it, because convergence says everything must be inside, so not true meaning whatever stage I give you I can find at least one point after that stage, so that it goes out of that window, it is something like in that pendulum, if after sometime I give a kick, I keep on giving a kick, so I can find always a window it will go out, it will not be convergent, something like that should happen, so not convergent, so let us look at that example that minus 1 to the power n divided by, sorry not divided by minus 1 to the power n is not convergent, so case 1 let, so this is my a n let limit minus 1 to the power n n going to infinity be equal to l, be equal to say plus 1, so let us assume, so if this is plus 1, this is 0, I know the sequence goes to minus 1 whatever stage, if the stage is even then at the odd stage next odd stage it will come out, it will go to minus 1, so positively it will go out of this kind of a window, so I have to select there exists an epsilon, so find an epsilon say that this is 1 minus epsilon 1 plus epsilon that is why I said let epsilon bigger than 0 be such that 1 minus epsilon is bigger than 0, then for every n not there exists n odd such that minus 1 to the power n does not belong to l minus epsilon n l plus epsilon, it will go out, so it will be minus 1 somewhere here at the odd stage that is outside, not only once it goes out whatever place I give you after that I take any odd stage it will go out, so this cannot be plus 1 cannot be the limit, similarly minus 1 can be the limit because then I will go to even, can something in between be the limit, this is 0, this is minus 1, this is plus 1, can this l be the limit, so obviously again if I take a window like this if I take a window like this then no n is going to be inside it, because n is only plus 1 or minus 1, so for this case I will choose minus 1 less than l minus epsilon less than l plus epsilon less than 1, so I choose for this kind of epsilon, so this is going to be my window, so nothing is going to be inside, leave aside something going out, so this cannot be the limit, so neither plus 1 can be the limit, neither minus 1 can be the limit, nor anything other than plus and minus 1 can be the limit that proves the sequence does not converge, so this is how we will write a proof of something not converging, right, is it time to stop, I do not know what is, what time we are supposed to be having lecture, what is the time now, 12.30, so let us stop here, so what we have done today is I have tried to give you some indication of why rationals are not good enough, why we need reals, what are reals, what are the important properties of reals which distinguish them from the rationals that is namely one completeness property and you look at the sizes, rationals are countable in infinite, reals are uncountable in number, that is something, then we looked at the why we need sequences, right and how do we analyze what we are interested in, when n becomes large what happens to the sequence n, we define the notion of convergence of a sequence, okay, let us stop.