 Hello, and welcome to a second example of proof by contradiction. Let's recap what we learned in the first video about proof by contradiction. This is a proof technique that's based on the fact that a statement and its negation have opposite truth values. So if I wanted to prove a statement to be true, I could prove its negation to be false. So to prove a statement, say, s, to be true using a proof by contradiction, I first assume that the negation of s is true. This is a temporary assumption that we will end up rejecting in the end. Then we use valid mathematical and logical steps, as well as facts and definitions, to work forward from this assumption, restating not s and exploring the logical consequences of assuming that not s is true. Eventually, we'll arrive at a contradiction to a known fact or a situation where two opposed facts must be true at the same time. This forces us to reject the assumption that s is not true, and so not s must be false. That means s, the original statement, must be true. So let's put this to work in another example. The theorem says that for all positive distinct real numbers x and y, we have that the square root of x, y is less than x plus y divided by 2. Just for your information, the two quantities here are two different kinds of averages. The quantity on the right is the regular average or what's called the arithmetic mean of x and y. The quantity on the left is called the geometric mean of x and y. Both of these kinds of averages show up in different places in mathematics and this theorem says that the geometric mean is always less than the arithmetic mean. Note the assumptions of this theorem especially, that x and y are positive real numbers and are distinct, which means that they are different from each other. They are not equal to each other. Now, proving inequalities can sometimes be tricky. It's easy to want to prove an inequality using a backwards proof. That is by starting by assuming that the inequality to be true and then working backwards. But we've seen that backwards proofs are logically flawed because we can't assume the result we're trying to prove. So a proof by contradiction often works really well in situations like these with inequalities because we actually get to assume something. We'll see this in just a moment. So to prove this by contradiction, we're going to first assume the negation of the statement we're trying to prove. The theorem statement is universally quantified. So the negation would say, there exist positive distinct real numbers x and y such that the inequality goes in the other direction. That square root of xy is greater than or equal to x plus y over two. Now notice, this is an inequality that we may use and we may work with. In fact, on the next slide we'll begin to do some valid mathematical steps to see what happens if we make this assumption. But realize this is not a backwards proof. I'm not assuming the result. I'm simply assuming the negation of the statement I'm trying to prove. So it's okay to start here and work forward. So here's the inequality we're assuming again. We can simplify this by squaring both sides. Now since x and y are positive, all the expressions you see here are positive. So we don't have to worry about the inequality changing directions when we square. Squaring both sides gives this expression where the numerator on the right happens because of the FOIL method. So let's clear the fraction and then subtract 4xy from both sides in an attempt to get all the variable quantities on one side. Why do we do that? It's just easier to work with algebra expressions if all the variable stuff is on one side. So we now have this inequality here. But notice, this is actually a perfect square. It's the square of x minus y. So we can factor it down into this form. Now so far all the steps we've done have been reasonable and valid math steps. But here we arrive at a contradiction. Now why is this a contradiction? Well, the expression on the right is a perfect square. So it's got to be greater than or equal to zero. But here I have it less than or equal to zero. Now this perfect square cannot possibly be negative. There is a possibility it could be zero, but that's actually ruled out here. It can't be zero in our case because x and y were assumed to be distinct real numbers. So x minus y isn't zero because x and y are different. So here we have an expression that is, we arrive at it being less than or equal to zero. But for basic algebra reasons, we cannot have it less than or equal to zero. So those two things cannot coexist, we're at an impasse or a contradiction. So what got us to this impasse? It was the assumption that there are distinct positive real numbers such that square root of x, y was greater than or equal to x plus y over two. It was by assuming the negation of our statement. That assumption must be wrong. So it must be the case that the negation of the statement is false. It must be the case that the original statement is true. Therefore, square root of x, y must be strictly less than x plus y divided by two. And that is the end of the proof. So there's another proof by contradiction. And I think it shows the power of the technique and its usefulness especially in working with things like inequalities. Thanks for watching.