So I've been trying to find a solution for this all afternoon, but haven't found a good place to start:
Prove that if $f:[a,b]\to\mathbf{R}^+$ is a continuous function with maximum value $M$, then
$$
\ \lim_{n\to\infty}\left(\int_a^b f(x)^n\,dx\right)^{1/n} = M
$$
Here are some of the paths I've considered, though none have been very successful:
(1) Considering the sequence of functions for all increasing integer $n$ and trying to show that the sequence converges. We've had plenty of work on converging sequences, but with the integral expression, I am not sure how to simplify.
(2) Showing that that sequence is increasing (again, how?) and then showing there to be a supremum at $M$. I'm not sure how the maximum of the function arrives in this problem.
(3) Mean value theorems for integrals
If anyone could give me a solid place to start or perhaps point me to a place where this question has been asked before (I can't seem to find it), I would be very grateful.
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