real analysis - Infinity norm related to $L_p(X)$ norm on finite measure space $X$.

Let $(X, \cal M, \mu)$ be a measure space with $\mu(X)<\infty$. Why

$$\begin{align} \lim_{p\to\infty}\|f\|_p=\|f\|_\infty \end{align}$$
for all $f\in L_\infty$?