I would like to show that: "every hereditary subalgebra $U$ of a $C^*$-algebra $C_0(X)$ for a locally compact Hausdorff Space $X$ has the form $J_E := \{f \in C_0(X) : f|_E=0 \}$ for a closed subset $E$ of $X$."

Any idea about how to show it concretely? (maybe using the Stone–Weierstrass Theorem?)