 Hi, I'm Zor. Welcome to a new Zor education. I would like to prove a very, very simple theorem which is usually called the squeeze theorem. It's about function limits. And obviously this is part of the advanced course of mathematics for teenagers and high school students presented on Unizor.com. And I do suggest you to watch this lecture from this website because every lecture and this one has very detailed explanation. It's basically like reading your textbook. So it's a live textbook, so to speak. Now, this theorem is very, very important and very, very simple. So it will be a very short lecture. Anyway, here is what it is. First of all about the name. It's called squeeze theorem. And there are some other names. The sandwich theorem, the pinching theorem and also much more colorful name which I used to know when I learned this theorem the first time. It's the theorem about two policemen and a drunk man. So here is what it is. Consider you have certain function and this function is always between two other functions. So there is one function which is above it and one function which is below it. So there is always this type of inequality. Now, let's consider that the smaller and the bigger function are all coming to a point at certain x equals to r and the point is, let's say, l. So whenever you are coming along this function to x equals r or this function to x equals r, we will actually have exactly the same limit on both functions. This one and this one both are going to limit l. So the theorem states that in this case this one which is in between also goes. So that's actually where the name two policemen and a drunk comes from. Because if they go to, let's say, some kind of police prison, he has no other choice because they are on both sides of him or a sandwich theorem or a squeeze theorem. So whatever you want to call it, more traditional is squeeze theorem, at least in American schools. All right, so now there are certain variations of this theorem. Now this is the variation which I'm going to prove. I'm not going to prove variation when instead of x equals to r, there is some kind of an infinity, which means if these two upper and lower bound functions are going at infinity, going to some limit, let's say this is level l. So this function goes to level l and this function goes to level l as x goes to infinity. Then again g of x also will go to the same limit l. That's one of the variations of this when the argument goes to infinity. Obviously plus infinity or minus infinity, it doesn't matter. Now another variation is when l is infinity, which means if these two functions are infinitely decreasing, let's say, then this one, which is in between, also will infinitely decrease or infinitely increasing. So that's when the l actually is, non-rigoriously speaking, equal in infinity. So I'm going to prove it in a very simple case, which is when l and r are normal real numbers. In case of infinity, it's just a very, very simple variation, which I'm not going to spend your time. I mean, it would be probably a very nice exercise if you can just prove it for yourself. But it's really a very, very simple change. So let's just do it in this particular case when I have some kind of a number here. So x goes to, let me put it this way, x goes to r and both of them and therefore the one in between goes to r. It goes to l and this is l. Alright, so how can I prove it? Well, let's use the epsilon-delta definition of the limit. Now, what does it mean that function f at x goes to l if x tends to r? It means the following. For any positive epsilon, however small, there should be some kind of a delta neighborhood of the point r, such that as long as my x is within delta neighborhood of r, immediately follows from this that f at x would be within epsilon neighborhood of l. So for any, however, small neighborhood of l, I can always find such neighborhood of r, so as long as argument is close to r, closer than delta, then my function will be closer to l. That's what basically definition of the limit is. Okay, so I know that. I also know exactly the same thing about h, right? So let's call this epsilon-1, that's for f. And now I can say exactly the same that there is a delta 2 such that if x is in the delta 2 neighborhood of r, then the function h also would be within epsilon neighborhood of l. So for any epsilon, no matter how small it is, I can always find delta 1 and delta 2 such that this is from this follows this and from this follows this. Now, very simple thing. Let's have delta equals to minimum of delta 1 and delta 2. What happens? Well, if delta is the minimum, then both are true. So we found such a delta when both of these are true. Now, let's just think about graphically. This is some number and this is some number and l is some number. Let's consider this is my number. So this is my l. What does it mean that f at x minus l less than epsilon? Well, if this is my l minus epsilon and this is l plus epsilon, so it means that f at x is somewhere here and h at x is also somewhere here. Let's say, yeah. And we always know that this is true. So in between them g must be somewhere in between, right? So that's kind of a graphical explanation. A little bit maybe less graphical would be the following purely algebraic kind of a logic. So from this, we know that f at x less than l plus epsilon and greater than l minus epsilon, right? From this, we know that h at x less than... And therefore, g at x is less than h at x less than l plus epsilon, right? But g of x is greater than f at x and f at x is greater than l minus epsilon. So we got again the same thing from l minus to l plus epsilon would be our g of x. So for this epsilon, whatever the epsilon we can choose, however small, we can always find some delta that this is true. We have found this delta that g of x is satisfying this inequality, which is absolutely equivalent to the fact that g of x minus l less than epsilon, which means g of x is within an epsilon neighborhood of l. So l is the limit by definition of the limits. So this is a very, very simple proof. This is just one-liner practical. I was talking a lot and I was drawing some graphs, but it's really one-liner proof. With infinity, either r is infinity, which means when x is infinitely growing or infinitely decreasing, or when l is infinitely growing or infinitely decreasing. So l is infinity, so to speak. It's really absolutely trivial and repetition of this. I think it would be a great exercise just for you. And you can send it to me as a proof and I will just publish it on my website if you want to, with proper attribution, of course. So anyway, that was my very short proof of a very, very important theorem because lots of very, very useful things in analysis of the functions and limits, function limits, can be proven using this technique. So if we find some kind of a lower bound for a function and higher bound and we can prove that both of them are going to the same limit, then whatever is in between will also go to this limit. That's how we can prove some statements about the limit of the g of x. All right, but that will be in the next lecture. I will definitely use it. That's it for today. Thank you very much and good luck.