Proposition 6.3.2.label Let $\frak{m}$ be a proper regular ideal of an abelian Banach algebra $A$. If $e$ is an identity modulo $\frak{m}$ then we have
\begin{align*}\INF{x\in\frak{m}}\norm{e-x}{}\geq 1\end{align*}
Proof. Suppose $\norm{e-x}{}<1$ for some $x\in\frak{m}$ then the power series $y=\sums{\infty}{n=1}(e-x)^{n}$. Since $(e-x)y=\sums{\infty}{n=2}(e-x)^{n}$, we have
\begin{align*}y=(e-x)y+e-x=ey-xy+e-x\end{align*}
hence $e=y-ey+xy+x=-(ey-y)+xy+x$ thus $e\in \frak{m}$. Lastly $\fall a\in\A$, $ea-a\in\frak{m}$ so $a\in\frak{m}$ contradicting $\frak{m}$ is proper.$\square$