Solution
= Solution
Because $A$ is a closed subalgebra of a <Banach algebra>, it is itself complete. If $\|a-1\|<1$, the <Neumann series> converges in $A$:
$$
\boxed{a^{-1}=\sum_{n=0}^{\infty}(1-a)^n.}
$$
Multiplying the partial sums by $a$ on either side gives $1-(1-a)^{N+1}$, which tends to $1$. Thus every such $a$ is invertible in $A$, and \b[$1$ is an interior point of $G(A)$].