Proof. Suppose $\spec{A}{x}=\emp$ for some $x\in A$. Using notation as in Proposition 7.3.6, the function $f_{\p}(\lam)$ is holomorphic everywhere; hence it is entire. But $\limit{\lam\to\infty}f_{\p}(\lam)=0$ so $f_{\p}$ is identically zero by Liouville’s Theorem, which means that $\inner{f(\lam)}{\p}\equiv 0$ for every $\p\in A^{*}$. Hence $f(\lam)\equiv 0$ contradicting $f(\lam)=(x-\lam)\inv$.$\square$