I'm currently looking at a proof in Royden's Real Analysis which applies Chebyshev's inequality in a way I'm not quite understanding. In particular, $$\int_{\{x\in A\, \mid\, f(x) \geq c\} } f \leq \frac{1}{c} \int_E f$$ where $f$ is nonnegative and integrable over $E$, $A\subseteq E$ is measurable, and $c>0$.
Note: This question has been posted before and an alternative proof of the claim was given, but there was no explanation for the above inequality.