Problem: Suppose that $X \subset \mathbb{R}^n$ is compact. Let $Y = \{y_1,...,y_p\}$ be a finite subset of $X$. Let $\mathscr{P} = \{p \in \mathbb{P}[x_1,...,x_n]: p(y_i) = 0$ for all $p\}$ (the collection of polynomials that vanish on $Y$). Show that $\mathscr{P}$ is dense on $I = \{f \in C_\mathbb{R}(X): f|_Y = 0\}$.
I'm not quite sure how to proceed with this problem. Is there any suggestion or hint?
EDIT: Dense wrt to the sup norm.
Hint: Suppose $Y$ is just one point, say $Y=\{a\}.$ By Stone-Weierstrass there exists a sequence $p_m$ of polynomials on $\mathbb R^n$ that converges uniformly to $f$ on $X.$ What can you say about
$$f(y)-[p_m(y)-p_m(a)]?$$