 For example, you can prove these properties of natural numbers just by assuming what I have said. For example, there is no natural number between n minus 1 and n. You can prove it actually using those properties of a field, right, okay. So let us assume. So now I want to state that crucial property for reals which distinguishes it from the rational. So for that, we are going to call a set to be a bounded, a is a subset of real number. We say a is bounded above. If there is a number, I have called it as alpha, say that alpha is bigger than or equal to every element of a. There is an order, right. Given any element a, I can compare it with alpha. So we want alpha to be bigger than or equal to every element of the set a, then we say alpha is an upper bound for the set a. We say a is bounded above. If all elements are less than or equal to something and that something is called an upper bound. Now you have got upper bounds. There could be more than 1, right. There could be more than 1 upper bound, right. For example, for the number 1, 2 is an upper bound, 3 is an upper bound, 4 is an upper bound, right. Because we know all successors are bigger than the number itself, right, the property. The smallest of these upper bounds, if it exists, is called least upper bound for that set, right. So let us define what is called the least upper bound for a set. It is a number which is an upper bound first of all. We are looking at the smallest of the upper bounds. Among the upper bounds, look at which is smallest one. If it exists, then we say that number is least upper bound of the set a. Now question is, can there be more than 1 least upper bound? No, because it is an upper bound and it has to be, right, so bigger than less than. So it has to be equal, right. So upper bound is always, if it exists, it is unique, right. So that is why we say the least upper bound normally, right. So the upper bound, if it is least, it is unique. So it is unique whenever it exists. Why to say upper only, right? We can also look at something smaller. So you can look at what are called sets which are bounded below. If there is a number, alpha says that alpha is less than or equal to a for every a, the element of the set a, right. All the elements of a are bigger than that element alpha, then alpha is called lower bound, okay. And now what you should be looking at? Are many lower bounds possible, the biggest of the lower bounds? So that is called the greatest lower bound for a set, if it exists, okay. So it is an lower bound and it is the greatest of them all. So that is called the greatest lower bound, okay. Now question is given a set a, subset of the real line. Can you say always that the set a may not be bounded? For example, look at natural numbers. Is it bounded? It is not because given any, if there is a bound something, we say that look at a integer and you can go on increasing it, right. So it becomes unbounded kind of a thing, right. So here is one example, let us look at all x between a and b. a and b are two real numbers fixed and let us look at the set of all real numbers so that a is less than x less than p, okay. By the construction of that set itself, the number a is a lower bound because every element of the set is bigger than a. b is a upper bound, right. Can you say that there is nothing bigger than a can be a lower bound? So try to prove it. Take it as an exercise, try to prove that if I give you a as the set, a less than x less than b, a and b are fixed, then a the set has got greatest lower bound namely a. It has least upper bound namely b. So try to, no, no, I will discuss it in the tutorial session, okay because I want everybody to think. You see the idea is not to get an answer from one person only. The idea should be that everybody should have time to think about it, analyze and see whether they are able to write down a proof of this fact or not that this has got a lower bound. For example a is a lower bound and actually that is the largest of the lower bounds, okay, right. So let me skip the argument that I wanted to give by saying that this is so. n is unbounded, you can prove these kind of things and you can prove what is called the Archimedean property of real numbers and that means what? That essentially says that for every x in R there exist a number nx, right, a integer, a natural number nx which depends on x such that nx is bigger than x. You will always cross over, unbounded is same as saying, right. Saying it is, if x is not a barrier, you can go beyond x by some natural number, right. And the same as saying a is unbounded, both are equivalent ways of saying the same thing, okay. I have not defined what is the sequence but essentially it says if you look at one over natural numbers intuitively it becomes smaller n, smaller n comes as close to 0 as you want it. We will make this precise slowly, okay. What does that mean? So this thing will make it precise, soon. Properties of natural numbers, okay. So here is what is called the order completeness property of real number. Every non-empty subset, right, of real numbers that is bounded above has a least upper bound. If there is a bound for it, for a set, it is bounded then there must be a smallest of the bounds possible. Intuitively it looks okay, it should be so. But Greeks discovered that that is not possible always. You can look at all rationals less than such that r square is less than 2, right. Saying that there is no rational whose square is equal to 2 is precisely almost saying that the set of all rational numbers r such that r square is strictly less than 2 is bounded above but does not have a least upper bound. One can prove that. If you want to know a proof of that, I will not be doing this proof. You can read this theorem in the book called From Numbers to Analysis, okay. We have a copy of that book in the library. So look at, just for your knowledge sake if you want to look at this theorem, right, it says that you can have subsets of rational numbers which are bounded above but they do not have a greatest lower bound as a rational number. And the completeness property says that if you treat any set whether rational or not and if it is bounded then it will have a least upper bound and that will be a real number, right. The set may be of rationals but the least upper bound will be always there, sometimes it may be of rational, sometimes it may be of rational we do not know but every non-empty subset of real numbers which is bounded above has got least upper bound. And almost a parallel way of saying that is if a set of real numbers is bounded below because if a set of is bounded above, if you look at negatives of those numbers that set will be bounded below from up to down it will always go by negative, right. So saying upper bound, negative lower bound, greatest least upper bound, greatest lower bound. So equivalent statement of this is that every non-empty subset of real numbers if it is bounded below it has got greatest lower bound. Both are equivalent statements and real numbers have got this property either of it and hence both of them. So this is what is called the completeness property of real numbers and precisely if you look at that set of rationals r square less than 2 and the least upper bound of that is precisely a real number whose square is equal to 2, one can prove that. So that will show the existence of roots of numbers, right in reals which may not be possible in rationals. We will be giving another way of stating this completeness property a bit later which is more sort of intuitively obvious and that property which says actually I will explain the terms of that it says every monotonically increasing sequence of real numbers. I have not defined what is the sequence, we will do that and we will say then that every monotonically increasing sequence of real numbers that is bounded above is convergent. So we will explain all the terms of this statement and then say what it means, right later on when I do what is the notion of a sequence. So completeness property of real numbers can be described in two different ways. One is least upper bound property or the greatest lower bound property, right that every non-empty subset of real numbers which is bounded above must have a above the least upper bound and every non-empty subset of real numbers which is bounded below must have a greatest lower bound equivalently will show it that every monotonically increasing sequence of real numbers if it is bounded above it must converge. We will say what are these terms and similarly every monotonically decreasing sequence of real numbers if it is bounded below it must converge. So those will prove it and show how is that true, okay. So here is something which our school teaches us and we start believing in that that all real numbers can be put on a line. We said that Greeks discovered that there is a point on the line which is not occupied by a any rational number namely the diagonal of a right angle triangle with sides 1, right. And later on so that is a kind of a if you put all rationales on the horizontal line they go and sit according to their size and positive negative order and everything but they leave gaps on the line. Those gaps are precisely the gaps filled by the numbers which we call now as irrational numbers, right. For example square root 2, square root 3 these are not the only numbers which are not filled. For example the number there is a number called pi and if you ask anybody what is pi, right the answer given is it is the area of the unit circle or it is the ratio of the diameter to the ratio of the circumference to the diameter of the circle. At least I do not know a school teacher who knows a proof of that theorem. I do not know anyone of you know that proof of that theorem or not that if I give you any circle given any circle look at the circumference and ratio of the circumference to the diameter that ratio always is the same quantity and that quantity we call it as pi. Try to discover a proof of that theorem. Now we have internet with you, okay. So go to internet try to search for a proof of the fact. No school book gives you the proof. Every school kid cramps this statement. Every teacher of school level assumes this statement blindly without even bothering whether it is true or false. There is none of them know a mathematical proof and unfortunately none of our books give a reference to that you can find a proof in this book at this place. Nobody gives and everybody believes that fact. So that is the way we study mathematics. You should not believe anything that is written in a book blindly or what Google says blindly, right. Nowadays everybody thinks what Google says is correct but who is Google? We are Google, right. We are putting up documents there. We may be making a mistake. So do not believe always test whether it is true or false. So try to find a proof of that fact that pi what we call as the ratio of the circumference to the diameter is a number on the number line. It is a real number. It is a ratio. So it should be a number. Is it rational or irrational? So ask for what is the proof. Why it is irrational? It is nothing of the type of square root of something. It is something totally new, right. So it is a irrational number. It is a theorem which is not very easy to prove. It requires some more, it is not as simple as saying there is no number of square is equal to 2. It is a bit more difficult there to prove. Anyway, so one can represent numbers on the line and no gaps are left. That means it says that there is a 1 to 1 correspondence between the numbers, real numbers and the points on the. So that is a geometric representation of the numbers, real numbers. Anyway there are some more properties one can prove namely between n and n plus 1, okay. There is a unique integer n such that given any number x, real number it must lie between n and n plus 1 for some n, right. Less than or equal to n strictly less than. And that n you call as the integral part of that number. That is the beginning of the story of decimal representation. That is where the decimal representation starts. And here is another important property which will not prove between any two real numbers. There is a rational number. Rational numbers are everywhere on the line. Given any two there is a rational number, right. But still there are many gaps left out. So it is an interesting question to ask how many rationales are there, how many points are there on the line, how many irrationals are there. You can think of this question. Can I sort of compare the size of rationales with the size of irrationals with the size of their union is the whole anyway, right. So what is the comparison? Again we will not be doing this but it is very interesting to know that there are as many rationales as there are points on the natural numbers. What do you mean by saying as many? Both are infinite things, right. So how do you compare infinities? Same way that we compute. How do you say there are 34 students in the class? Pick up one, number one, roll number one, roll number two, roll number one to one correspondence. So given any two sets you say they have same number of elements. You do not say number now. You say they have the same cardinality if there is a one to one on two map between the two. You can associate this with this, this with this and one to one on two. So there is a one to one on two map between natural numbers and rational numbers and one can prove there is no one to one on two map between the rationales and reals. Rationales are inside the reals, right. They are dense. They are everywhere as far as order is concerned but if you try to count, if you find out the size, the size of rationales is much, much smaller. That means there is a copy of rationales sitting inside reals of course but what is left out is much, much bigger. So one says rationales are countably infinite and reals are uncountably infinite. We would not be going much into these things. We just to make you understand the relation between rationales and reals. Rationales are dense. Everywhere as far as order is concerned, their size is much, much smaller as far as reals are concerned. So that is the relation between the two. So this construction as I said of real numbers was obtained by Richard Dedekind in 1858 which is not very far away, right. 1858 is only 150 years back, right. So you see after the Greeks it took almost 2000 years to define what is the real number, okay. And independently it was done by a mathematician called George Kent in 1872. Both the approaches of construction are different. Both start with a copy of rational numbers and both construct a new object which is a complete ordered field which has a copy of rationales sitting inside it as a dense set. So nothing more than that. And the geometric realization that Greeks believed every number must be a point on the line that was realized by Wallace as a mathematician John Wallace who gave there is a one-to-one correspondence between set of real numbers and the geometric object of points. So dream of Greeks was realized by the work of Wallace, right. So whenever we want to give examples on the real line which help us to understand we take real line as real numbers as a line, real numbers as a set, real numbers as a line that is only for visualization purpose to visualize some things, right that helps, okay. So that is the real numbers. Now let us come to what is called sequences. Why do you think one should be studying sequences of anything? Are the sequences around us? They are everywhere. You go and buy one to buy a ticket of something, you stand in a queue. What you are doing? In English you will say you are forming a queue but I will say you are forming a sequence. And in that whenever somebody tries to come ahead of you, you say a queue is behind, right there is an order in this queue. I am the first, he is the second, he is the third and you are the last, you go in the end. So a ordered collection of objects, an order, a collection of all objects, it is ordered meaning there is order to every object, the first object, second or third and so on is called a sequence of that objects. It could be of anything, sequence of human beings, sequence of numbers or anything, right. So here are some examples which are very, which goes back to what we were discussing. Let us see if I can show you. So let us just look at this clip. Blue is the circle area I want to find out and I am approximating it by putting in a rectilinear figure inside it because for rectilinear figures we know what are the areas I can compute and if I increase the number of sides, I am filling up more and more part of the circle. This is what Euclid had done. He tried to find out what is the area of the circle by this method by putting the regular n-gaons inside the circle and finding their areas. So what is this giving me? When I put a triangle inside, I get A1 area of the triangle. When I put a square inside it, I get another observation that is my second approximation A2, third, A5, 5-sided n, right pentagon, so A3, A4. So I am getting a sequence of observations and what I want to know about the sequence of the observations is eventually these n-gaons should fill up the circle, right. For n large enough, these approximations give me a good estimate of what I can call as the area of the circle. They do not help me to define yet I have not obtained what is the area of the circle, unit circle. I have only got approximations but I know that geometrically it is coming closer and closer to the object I want. So how do I formalize that notion that these things are coming closer and closer, right. So another way of doing that would be the following. So let me try to do that slightly better what Euclid had done. I am not better than what Euclid had done in slightly different way. So let us look at the unit circle. So let us, I start with a regular four-sided thing inside it that is a square. So that will give me, so that will cover this part of the area. So what is left out? So let me call this as E, Q, R and S. So what is left out is the sectors they are not covered. I want to cover them. So what I do is instead of a pentagon I do the following. I take the midpoints of this and join them. Now look at the area of this. How many sides are there? Eight sides of the octagon and this octagon not only covers the square actually it covers something more. These are also covered. These triangles are also covered now. So first iteration area of the square, second iteration A2 area of the octagon. Clearly it is bigger than the area of the circle. Geometrically it is clear. And what is to be my next step? From octagon I should go to 16 guns. So what I will do is I will write down, this is my, for the square less than, less than and geometrically it is quite clear these are, these numbers each one is strictly bigger than the previous one. And all these areas are inside a square which is circumscribing the circle. So all these numbers A4, A8, A16 all are increasing. Each one is bigger than the previous one but they do not go beyond something. So what do you think should happen to them? They should get cramped and cramped eventually and they all should come closer and closer to something and that is precisely the notion of convergence. So that is the completeness property of real numbers. It says a monotonically increasing sequence of real numbers which is bounded above must come closer to a value. So let us make this as a bit more precise so that we can go ahead. So a sequence of, so everybody understand what is the sequence of numbers? A sequence of numbers is the ordered collection of numbers that is the sequence first term A1, second term A2, third term A3 and so on. And this is the way we write it. A sequence is written as An, An is the nth term with this curly brackets and bigger than or equal to 1. This is not the same as the set An by the way. This notation is much different from this is the notation for saying you are looking at the An's as elements of a set. This is looking at the sequence An. So let me make it slightly more clear. These two are different things. For example, I can look at An as the population of India in the some year nth here. Let us look at minus 1 to the power n, the sequence. What is the first term n equal to 1 that is equal to minus 1, second term plus 1, third term minus 1. So as a set it is only two element set plus minus 1 but as a sequence it is minus 1 plus 1, minus 1 plus 1, minus 1 plus 1. So that is the difference between a sequence and sequence as a set. You can have for example this cos n pi. You can find out the values again. Is this the same as the previous sequence? Second and third, both are same? No? Think about it. n equal to 1, what is cos pi? So what is the first term there in the second one? What is the second term? I would not say anything more. For example, this is another sequence. So let us write down what is meaning that what happens to n when An becomes large and large. In the part of the circle thing we hope that the n goes n equal to 2 n goes, the increase and the bounded.