real analysis - p-norm and the sup norm

$f$ is a positive continuous function on a compact interval $[a,b]$. Determine the limit

$\lim_{n \to \infty}[\int_{a}^{b}f(x)^ndx]^{1/n}$.

For this question, isn't the limit just the sup norm of $f$? If it is, how to show it formally? (and why does $f$ have to be positive?)

Thanks