I proved the following theorem but I'd like to confirm the last part of my proof. Statement:
Let $A$ be a non-zero commutative $C^\ast$ algebra. Then $\varphi : A \to C_0 (\Omega(A))$ defined by $a \mapsto \widehat{a}$ (Gelfand representation) is an isometric $\ast$ isomorphism.
I had no trouble showing that it is an isometric $\ast$-homomorphism. Then to argue why it is surjective I want to use Stone Weierstrass to show that $\overline{\varphi (A) }= C_0(\Omega(A))$. So I argue as follows:
(i) $\varphi (A)$ vanishes nowhere: Let $\tau \in \Omega(A)$. Then $\tau$ is a character and therefore by definition non-zero hence there exists $a \in A$ with $\tau (a) \neq 0$. Hence $\widehat{a}(\tau) \neq 0$.
(ii) $\varphi (A)$ separates points: Let $\tau, \tau' \in \Omega (A)$. Then $\tau - \tau'$ is also a character hence by definition non-zero. Hence there exists $a \in A$ with $(\tau - \tau')(a) \neq 0$. Hence $\widehat{a}(\tau) \neq \widehat{a}(\tau')$.
Is this really correct? Saying something follows from the definition seems a little too easy.