If $(x_n)$ converges to a non-invertible element, then $\lim \| x_n^{-1} \| = \infty .$

Question

Let $(x_n)$ be a sequence of invertible elements in a Banach algebra $A$ with identity $e$ converging to a non-invertible element $x.$ Prove that $\lim \| x_n^{-1} \| = \infty .$

My attempt: I tried proving it by contradiction. If $\lim \| x_n^{-1} \| \neq \infty,$ then we may assume that there exists $C>0$ such that $\|x_n^{-1}\|\leq C$ for all $n \in \mathbb N.$

Then I tried to get a contradiction to the fact that $x$ is not invertible by showing $\|e-x\|<1,$ but I couldn't show this.

Am I on the right track? Any hints are appreciated.

Answer 1

Hint: Show that $\|e-x_n^{-1}x\| <1$ for some $n.$