Hello all! I was recently reading a proof of the AM-GM inequality by George Polya in my book titled "The Cauchy-Schwarz Master Class" by professor J. Michael Steele. I think it's an extraordinary, beautiful proof, but there's one step that I don't quite understand. I'll post a relatively quick proof, pointing out my area of confusion when I get there.
So, we want to prove that for a sequence of reals with and a sequence of reals with such that , that
Polya's proof begins with the observation that which can easily be seen graphically with equality occurring only at . If we make the change of variables , our initial observation becomes . Applying this to our sequence we get
now, we can see that however, we can also see from one of our initial observation that which is the same bound we just found for . So we could say
So we have related and by inequality, but we have not separated them. Now we look closer at the case where and are equal to the expression on the right. The idea of "normalization" then comes to mind. Normalization was discussed earlier in the book, but What Steele does is the following:
Define a new sequence with where we have where we have
Applying our earlier bound for for our new variables , we get Now, Steele implies that and it follows that
Now from here, showing that the AM-GM inequality holds is easy. The only place that brings me confusion is the purpose of introducing and defining the sequence in the way that Steele does. If someone could help me out with this, it would be greatly appreciated! Consider this a recommendation for "The Cauchy-Schwarz Master Class," it's a fantastic book and you'd all love it!