This is from a course in real analysis. Let $f:G\rightarrow \mathbb{R}$ be a continuous bounded function and $G\subset\mathbb{R}^n$ an open bounded set. Prove that for every compact set $K\subset G$ there is a series of polynomial that converge uniformly on $K$.
I can't use Stone-Weierstrass theorem here . I can use the fact that polynomials are dense in $L_p$. Any ideas?