Integral of a continuous function

Question

It is given a continuous function $f$ on $[0,1]$. Assuming that $$\int_{0}^{1} x^n f(x) \ dx = 0$$ for every $n\in\mathbb{N}$, how to prove that $f(x) = 0$ on $[0,1]$?

Answer 1

approximate the function by sequence of polynomials. Weierstrass' Theorem.

Answer 2

Assuming that $f(x)\not\equiv 0$, then $\left|f(x)\right|\geq c>0$ over some interval $I\subset [0,1]$. The trick now is to consider some Bernstein polynomial that is concentrated over $I$, or some good polynomial approximation of $\mathbb{1}_I$, and integrate it against $f(x)$. By the assumptions, such integral must be zero, but it should be also non-zero by the concentration (or approximation) argument, so assuming $f(x)\not\equiv 0$ leads to a contradiction.