real analysis - Convergence from $L^p$ to $L^infty$

If $f$ is a function such that $f \in L^\infty \cap L^ {p_0}$ where $L^\infty$ is the space of essentially bounded functions and $0 < p_0 < \infty$. Show that $|| f|| _{L^p} \to ||f || _{L^\infty}$ as $p \to \infty$. Where $|| f||_{L^\infty}$ is the least $M \in R$ such that $|f(x)| \le M$ for almost every $x \in X$.

The hint says to use the monotone convergence theorem, but i can't even see any pointwise convergence of functions.
Any help is appreciated.

Answer

Hint: Let $M\lt\|f\|_{L^\infty}$ and consider
$$\int_{E_M}\left|\frac{f(x)}{M}\right|^p\,\mathrm{d}x$$
where $E_M=\{x:|f(x)|\gt M\}$. I believe the Monotone Convergence Theorem works here.

Further Hint: $M\lt\|f\|_{L^\infty}$ implies $E_M$ has positive measure. On $E_M$, $\left|\frac{f(x)}{M}\right|^p$ tends to $\infty$ pointwise. MCT says that for some $p$, the integral above exceeds $1$.