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If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?
I'm trying to learn some analysis on my own and I stumbled upon the following proposition.
Suppose $ f:[a,b]\rightarrow \mathbb{R} $ is continuous. Suppose $ f(x) \geq 0 \space$ for all $ x \in [a,b] $. Let $ M=sup\{f(x):x\in[a,b]\} $
Then the sequence $ \{[\int_a^b[f(x)]^n\,dx]^{1/n}\}_{n=1}^\infty $ converges to M.
I'm really not sure where to begin in proving this.