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**Homework Statement**

Let f: [a,b] -> R be continuous such that f(x) ≥ 0 for all x in [a,b]. Show that

[tex]\lim_{n \to \infty} \left( \int_a^b (f(x))^n \, dx \right)^{1/n} = \max\{f(x) : x \in [a,b]\}.[/tex]

**Relevant equations**

The fundamental theorem of calculus and its corollaries.

**The attempt at a solution**

It is easy to show that the limit is less than or equal to the max of f. This doesn't rely on using the fact that f ≥ 0, so somehow this extra piece of info. turns 'less than or equal' into 'equal', but I have failed to determine why. Any tips?