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$

Question

Let $f$ be a nonnegative continuous function on $[a,b].$ Let $M=\max \{f(x):x\in[a,b]\}$. Show that \begin{align} \lim\limits_{n\to \infty} \left[\int^{b}_{a}f^{n}(x)dx\right]^{\frac{1}{n}}=M\end{align}

MY TRIAL

My idea is to estimate and arrive at $\epsilon.$ \begin{align} \left|\left(\int^{b}_{a}f^{n}(x)dx\right)^{\frac{1}{n}}-M\right|&\leq \left|\int^{b}_{a}f(x)dx-M\right| \\&= \left|\int^{b}_{a}f(x)dx-\int^{b}_{a}\frac{M}{(a-b)}dx\right| \\&= \int^{b}_{a}\left|f(x)-\frac{1}{a-b}M\right|dx\end{align} I'm stuck here but I sense that the above idea of mine, sounds quite unscholarly of me but I could do nothing more? Please, are there any better explanations out there? Thanks!