Exercise 12, § 3

Let A be a unital Banach algebra and $(x_n)$ a sequence of invertible elements of A tending to an element $x\in A$. If the sequence $(\varrho (x^{-1}_n))$ is bounded, and if $x_nx=xx_n$ for all $n$, then $x$ is invertible. (According to the Cor. of the Prop. 8 of I, p. 38$,\varrho (1-x^{-1}_nx)\leqslant \varrho (x^{-1}_n)\varrho (x_n-x)$, hence $x^{-1}_nx$ is invertible for sufficiently large $n$.)