Let $\left(X,M,\mu\right)$ be a measure space. Suppose $f:X\to[0,\infty)$ is measurable and the set $A:=\left\{ x\in X\mid f\left(x\right)>1\right\}$ has positive measure. Show that $\lim_{p\to\infty}\int_{A}f^{p}d\mu=\infty$.
Hint: Let $A_{n}=:\left\{ x\in X\mid f\left(x\right)>1+\frac{1}{n}\right\}$.
Even with the hint, I still don't know where to start. Any help will be much appreciated.
The sequence {$A_n$} increases to $A$. So, $\mu(A_n) \leq \mu(A)$ for each $n$(By montonicity) . Since $\mu(A)< \infty$, $\mu(A_n)<\infty$. So,
$\int_{A} f^p d\mu \geq \int_{A_n} f^p d\mu > \int_{A_n} (1+ \dfrac{1}{n})^p d\mu=(1+ \dfrac{1}{n})^p \int_{A_n}1 d\mu = (1+ \dfrac{1}{n})^p \mu(A_n)$
Letting $p \to \infty$, the rightmost term apporaches $\infty$. Hence, $\int_{A} f^p d\mu \rightarrow \infty$ as $p \to \infty$