The question statement is as follows: Consider a continuous function $f : [a,b] \to \Re^+$ and $M_p = (\int_a^b (f(x))^p dx )^\frac{1}{p}$, prove that $\lim_{p \to +\infty} M_p = \max_{x\in[a,b]} f(x)$
After some thinking I've gotten to the result that $\lim_{p \to +\infty} log(M_p) = \lim_{p \to +\infty} \frac{\int_a^b log(f(x))(f(x))^p dx}{\int_a^b (f(x))^p dx }$ but I don't know how to move forward from here. Any help would be appreciated.
