Let $\Omega$ be a locally compact Hausdorff space, and suppose that the $C^{*}$-algebra $C_{0}(\Omega)$ is generated by a sequence of projections $(p_{n})_{n=1}^{\infty}$. Show that the hermitian element $h=\sum_{n=1}^{\infty}p_{n}/3^{n}$ generates $C_{0}(\Omega)$.
I think it is quite clear that the $C^{*}$-subalgebra generated by $h$ is in $C_{0}(\Omega)$ but I can't show equality.
For any positive linear functional $\tau$ on $C_0(\Omega)$, $$ \tau(h) \geq \frac{1}{3^n}\tau(p_n) \quad\forall n\in \mathbb{N} $$ Hence, if $\tau(h) = 0$, then $\tau = 0$. Conclude that $C^{\ast}(h) = C_0(\Omega)$