How do you show that if $x_1,\ldots,x_n$ are real numbers, then $|x_1|\ldots|x_n| \le |x_1|^2 + \cdots + |x_n|^2$. I tried using induction, but I'm guessing that's not the way to do it? Help would be appreciated.
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Let $x_k = 2$. Then $|x_1|\cdots|x_n| = 2^n$, but $|x_1|^2 + \cdots + |x_n|^2 = 4n$. Choose $n=5$, then $32>20$.
Maybe this would help to clear the original statement: http://en.wikipedia.org/wiki/Generalized_mean?