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Let $I$ be the interval $[a,b]$ and suppose that $f:I\to\mathbb{R}$ is continuous on $I$ with $f(x)\geq 0$ $\forall x\in I$. How can we show that the sequence $$\left(\int_a^b f(x)^n dx\right)^{1/n}$$ for $n\in\mathbb{N}$ converges to $M:=\sup\{f(x):x\in I\}$?

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marked as duplicate by L. F., Julien, Norbert, Amzoti, hardmath Apr 16 '13 at 2:42

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