threeder
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Homework Statement
Prove that
\frac{1}{n}\sum_{i=1}^n x_i\geq {(\prod_{i=1}^n x_i)}^{1/n}
for positive integers n and positive real numbers x_i
Homework Equations
There is also a hint. It states that it does not seem to be possible to prove it directly so you should prove it for n=2^m, m\geq 0. Nonetheless, the result should follow by proving the converse of the usual inductive step: if it hold for n=k+1, the it also does for n=k
The Attempt at a Solution
I don't really understand how should I proceed. I rewrote the inequality
\frac{1}{2^{k+1}}\sum_{i=0}^{2^{k+1}} x_i\geq {(\prod_{i=0}^{2^{k+1}} x_i)}^{2^{-k-1}}
but that's it. Can't seem to understand how can we proceed from the converse using the induction principe. Any hints would be apreciated. Thanks!
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