Integral inequality with $L^p$ norm

Question

Suppose $f(x)$ is p-th integrable, i.e. $\|f\|_p < \infty$ on some measurable set $A$. Let $A_N = \{x\in A \, : \, |f(x)|>N\}$. Is there any simple reason why the following inequality is true? $$ \int_{A_N}|f(x)|\, dx \leq \frac{1}{N^{p-1}}\|f\|_p^p$$ for $N \in \mathbb{N}, p >1$. There is a proof of this using Holder's inequality, but I am wondering whether there is some simple, quick proof that I am missing. Thanks for your help.

Answer 1

Recalling that $|f(x)|>N$ on $A_N$ we see that $\frac{||f||_p^p}{N^{p-1}}=\frac{1}{N^{p-1}}(\int |f(x)|^p) \geq \frac{1}{N^{p-1}}(\int_{A_N} |f(x)|^p) \geq \frac{1}{N^{p-1}}(\int_{A_N} |f(x)|N^{p-1}) = \int_{A_N} |f(x)|$