Convergence in C* algebra forces spectra to approach limit point's spectrum

Question

Let $A$ be a $C^*$ algebra with unity and let $x$ be a normal element and $(x_n) $ a sequence of normal elements converging to $x$. Also let $\Omega$ be a compact neighborhood of $\sigma(x) $. I am trying to prove that for large enough $n$ it is $\sigma(x_n) \subset\Omega$. I am not even sure if it is true and I dont know where to start. Any hints?

Answer 1

Normality isn't needed, this result holds for unital Banach algebras.

There is some $M>0$ such that $\|(x-\lambda)^{-1}\|<M$ for all $\lambda\in\mathbb C\setminus\Omega$. If $\|x_n-x\|<\frac{1}{M}$ and $\lambda\notin\Omega$, then $$x_n-\lambda=(x_n-x)+x-\lambda = (x-\lambda)\left[(x-\lambda)^{-1}(x_n-x)+1\right]$$ is invertible, since $\|(x-\lambda)^{-1}(x_n-x)\|<1$, and hence $\lambda\notin\sigma(x_n)$. Thus if $\|x_n-x\|<\frac{1}{M}$, then $\sigma(x_n)\subset\Omega$.