Corollary 33.2.4.label Let $A$ be a unital Banach algebra, $x \in A \setminus G(A)$, and $r > 0$, then $\normn{y^{-1}}_{A} > 1/r$ for all $y \in B(x, r) \cap G(A)$.
Proof. Let $y \in B(x, r)$, then $B(y, \normn{y^{-1}}_{A}^{-1}) \subset G(A)$ by Proposition 33.2.3. Since $x \not\in G(A)$, $r > \norm{x - y}_{A} \ge \normn{y^{-1}}_{A}^{-1}$, and $1/r < \normn{y^{-1}}_{A}$.$\square$
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