Let $E$ be the set of continuous functions $f$ from $I$ to $\mathbb{R}$ such that $f^2 w$ is integrable on $I$.
Show that, for all functions $f$ and $g$ in $E$, the product $fgw$ is integrable on $I$. You may use the inequality $\forall (a,b) \in \mathbb{R}^2, |ab| \leqslant \frac{1}{2}(a^2 + b^2)$, after justifying it.