These are two closely related question to prove
1)If $\phi\in C_0^{\infty}$,and $\int_{\mathbb{R}^n}x^{\alpha}\phi=0$ for all multiindices $\alpha$, where $\alpha=(\alpha_1,\cdots,\alpha_n),x^{\alpha}=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ then $\phi=0$
2)There exists $\phi$ in Schwartz class s.t. $\int_{\mathbb{R}^n}x^{\alpha}\phi=0$.
It is quite confusing because $C_{0}^{\infty}$ is dense in Schwartz class and I don't know how to prove these two statements.