Solution
= Solution
The given <moments> imply $\int_0^1 f(x)P(x)\,dx=0$ for every <polynomial> $P$. Choose <polynomials> $P_k$ converging uniformly to $f$ by the <Weierstrass approximation theorem>. Then
$$
\left|\int_0^1 f^2\right|=\left|\int_0^1 f(f-P_k)\right|\leq\|f-P_k\|_\infty\int_0^1|f|\longrightarrow0.
$$
A nonnegative <continuous function> with zero <integral> vanishes everywhere. Hence \b[$f=0$.]