real analysis - Limit of integral expression approaches maximum of function

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.