Let $||f||_2 = (\int_{r-1}^{r_2}|f|^2)^{1/2} $.
I have that
$\star \int_{r_1}^{r_2} |f \cdot g| \leq (\int_{r_1}^{r_2} |f|^2)^{1/2} \cdot (\int_{r_1}^{r_2} |g|^2)^{1/2}$
for continuous functions $f,g$ on real interval $[r_1,r_2]$, and I want to show that for continuous functions $f,g,h$ on $[r_1,r_2]$, we have $||f-h||_2 \leq ||f-g||_2 + ||g-h||_2$.
I have tried just proving this outright by looking at $||f-h||_2, ||f-g||_2, $ and $||g-h||_2$, but I can't see how $\star$ relates to this problem. I'd just appreciate some hints as to how this proof works. Thanks!
