= Inverse norm divergence at noninvertible boundary points
{title2=$a_n\to a\notin G(A)\Longrightarrow\|a_n^{-1}\|\to\infty$}
In a unital <Banach algebra>, if a subsequence of inverse norms remained bounded, $a=a_n[1+a_n^{-1}(a-a_n)]$ would be invertible for large subsequence indices by the <Neumann series>. This contradicts noninvertibility of $a$. If $A$ is a closed unital subalgebra of a larger <Banach algebra> $B$ and the $a_n$ lie in $G(A)$, then such a limit cannot be invertible in $B$ either: <continuity of inversion in a Banach algebra> would make their inverses converge in $B$ to $a^{-1}$, which closedness puts in $A$.
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