Let $f$ be a nonnegative continuous function on $[a,b]$, then $ \lim\limits_{n\to \infty} \left[\int^{b}_{a}f^{n}(x)dx\right]^{\frac{1}{n}}=M$
Prove that the sequence $\{v_n\}$ converges to the maximum value of $f$ on $[a,b]$