# Problem 40

Here is solution to #2. By Weierstrass Approximation Theorem there is a sequence of polynomials $p_n(x)$ converging uniformly to $f(x)$. But $\int_0^1 p_n(x)f(x) dx = 0$ (because if you write $p_n(x) = a_nx^n+...+a_1x+a_0$ it should be clear by hypothesis). But then $\lim \int_0^1 p_n(x)f(x) dx = \int_0^1 f^2(x) dx = 0 \implies f(x)=0$ (this is were we use uniform convergence). Q.E.D.