Q:

Prove that it is impossible to bound the integral of a product by the product of integrals

How to prove that there is no $c  \ge 0$ such that
$$
\Big|  \int_{0}^{1} f(x)g(x) dx  \Big|  \le c  \int_{0}^{1} |f(x)| dx   \cdot   \int_{0}^{1} |g(x)| dx
$$
where $f,g: [0,1] \to \mathbb{R}$ are continuous.

A:

Take $f(x)=x^n=g(x)$. Then there is no $c$ so that
$$ \frac{1}{2n+1} \leqslant \frac{c}{(n+1)^2} $$
for every positive $n$.

