AM - GM inequality proof

[anabaptist]

Hello, I am new to this forums and I need some help in proving the AM - GM - HM inequalities.

Can anyone please explain me the elementary proof by induction found here:
http://en.wikipedia.org/wiki/Inequa...geometric_means#Elementary_proof_by_induction ?

Our teacher requires us to know how to prove them, and I must fully understand it in order to learn it.

 

[JeffM]

Hello, I am new to this forums and I need some help in proving the AM - GM - HM inequalities.

Can anyone please explain me the elementary proof by induction found here:
http://en.wikipedia.org/wiki/Inequa...geometric_means#Elementary_proof_by_induction ?

Our teacher requires us to know how to prove them, and I must fully understand it in order to learn it.

Without knowing what step loses you, it is hard to know where to start. The third step is badly worded and highly condensed.

The proof starts by saying that the arithmetic mean and geometric mean are equal if the number of items being averaged is one. Clear.

The proof then says that if all the items being averaged are equal, then the arithmetic mean and geometric mean are equal to the items being averaged and so must also be equal to each other. Clear

So the case now being considered involves more than one item to be averaged, and not all of them equal to the arithmetic mean. The proof then says, "you may find." What is meant IN PART is that at least one item of the items being averaged MUST be greater than the arithmetic mean and at least one of the being averaged MUST be less than the arithmetic mean. Wrong modal verb. This is assumed without proof, but it is easy enough to prove by contradiction. (For me, the proof is spoiled by this failure to disclose a necessary lemma.) But what the "may find" ALSO means is that you can rearrange the terms of a sum or a product without any effect on the sum or product, and it places one of the greater items (there must be at least) and one of the lesser items (there must at least one) as the final items being added or multiplied. It is a "without loss of generality" statement.

That was the first place in this proof where I had to stop and ponder. Unless you tell us where you are stuck, I doubt any of us wants to try to rewrite a wiki article.

PS You did not say what your math background is. Do you understand proofs by induction in general?

 

[anabaptist]

Without knowing what step loses you, it is hard to know where to start. The third step is badly worded and highly condensed.

The proof starts by saying that the arithmetic mean and geometric mean are equal if the number of items being averaged is one. Clear.

The proof then says that if all the items being averaged are equal, then the arithmetic mean and geometric mean are equal to the items being averaged and so must also be equal to each other. Clear

So the case now being considered involves more than one item to be averaged, and not all of them equal to the arithmetic mean. The proof then says, "you may find." What is meant IN PART is that at least one item of the items being averaged MUST be greater than the arithmetic mean and at least one of the being averaged MUST be less than the arithmetic mean. Wrong modal verb. This is assumed without proof, but it is easy enough to prove by contradiction. (For me, the proof is spoiled by this failure to disclose a necessary lemma.) But what the "may find" ALSO means is that you can rearrange the terms of a sum or a product without any effect on the sum or product, and it places one of the greater items (there must be at least) and one of the lesser items (there must at least one) as the final items being added or multiplied. It is a "without loss of generality" statement.

That was the first place in this proof where I had to stop and ponder. Unless you tell us where you are stuck, I doubt any of us wants to try to rewrite a wiki article.

PS You did not say what your math background is. Do you understand proofs by induction in general?

Well, sorry for not fully introducing myself. Well, I am currently 3rd year high school, kind of "majoring" in maths (so I have algebra and linear algebra), and I am familiar with some of the concepts presented in the proof (on wikipedia), and even looked at the other proofs, but this is the one that I found to be easiest of all, but I really can't understand the logic of the last 3 lines. Some things are kind of too complex for a 17 yr old like me to understand. I really have a hard time understanding the proofs.

 

[JeffM]

Well, sorry for not fully introducing myself. Well, I am currently 3rd year high school, kind of "majoring" in maths (so I have algebra and linear algebra), and I am familiar with some of the concepts presented in the proof (on wikipedia), and even looked at the other proofs, but this is the one that I found to be easiest of all, but I really can't understand the logic of the last 3 lines. Some things are kind of too complex for a 17 yr old like me to understand. I really have a hard time understanding the proofs.

OK Let's work this way. You go line by line. Whenever you are not absolutely sure that you understand what a line means or where it comes from, identify the line so we can see if we can explicate it. I agree with you that the exposition is hardly a model of clarity.

What is important is to identify each line where you have any uncertainty before proceeding to any later line. Proofs are cumulative, and any vagueness in the beginning just gets magnified toward the end.

 

[anabaptist]

OK Let's work this way. You go line by line. Whenever you are not absolutely sure that you understand what a line means or where it comes from, identify the line so we can see if we can explicate it. I agree with you that the exposition is hardly a model of clarity.

What is important is to identify each line where you have any uncertainty before proceeding to any later line. Proofs are cumulative, and any vagueness in the beginning just gets magnified toward the end.

Okay, I get this one.

this one too.

Now to the third line.

I cannot clearly understand the relation here.

How does he come up with this one:

I understand the next 2 steps:

And the next 2 are obvious.