Proving the AMGM inequality by proving forwards

[twilit birch]

Can anyone explain why this method of proving forwards doesn’t work but the Cauchy induction does work?

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[novel bone]

you've only proven it for the case where c is (a+b)/2 though

like you have not actually proven three-variable am-gm

[twilit birch]

that's the problem it assumes that c is the mean of a and b

but for the cauchy proof it also assumes that the Xn term is the mean

[novel bone]

uh that's substituting a value for xn into the n-variable case and deriving the n-1 variable case from it

[twilit birch]

but isnt this term the AM of the previous terms

so once again it only proves the case where the final term happens to be the AM of the previous terms

[novel bone]

no but the final term doesn't exist

like you're explicitly reducing it down into the n-1 variable case

it's like if I wanna go from
x^2 + y^2 >= 0
to x^2 + 4 >= 0

I'm allowed to do that

but I can't say "set y=2, since x^2+4 >= 4 we have x^2 + y^2 >= 4"

[novel bone]

like have you noticed that the variable xn does not exist in the final line/conclusion of the statement

[hoary goblet]

...it literally states that it's assuming the AM-GM inequality holds for n terms. That is, for n arbitrary terms, in particular when the nth term is the arithmetic mean of the other terms. And then proving that the AM-GM inequality holds for n - 1 terms.

[twilit birch]

alright so say we try to prove 3-way AMGM from 4-way

assume a+b+c+d >= 4(abcd)^(1/4)

then let d = (a+b+c)/3

what happens if d isn't (a+b+c)/3

[novel bone]

then you get a different and equally valid case of the 4-way AMGM that is most likely going to be fucking useless for proving the 3-way case

like sure yeah you can do that and it’ll work but that’s entirely not the point

[novel bone]

please refer to this analogy if you are in the habit of understanding analogies

[twilit birch]

ohh wait
so letting d=(a+b+c)/3 is just one specific case of the 4-way that happens to reduce into the 3-way case nicely
and my original proof here assumes one case for a 3-way AMGM that isn't universally true

?

ok nah im just slow sry about that

[royal kettle]

Unable to parse the channel name

[twilit birch]

+close

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