trying to proof $f \in L_p, ||f||_r → ||f||_\infty$

Question

I've found a solution in a book (image) but I don't get this step... maybe if f and g are non negative, is $\int f.g≤\int f\int g$ a property? I don't remember this really. $0<p<\infty$ https://i.stack.imgur.com/tr9jI.png

Answer 1

The set $A=\{ |f| > \|f\|_{\infty}\}$ has measure zero so when integrating over space $X$, it suffices to integrate over $A^c$. On this space, we know $|f|\le \|f\|_{\infty}$ which you apply to the $|f|^{r-p}$ term.