continuous depence of the spectrum on elements Suppose $a_n \to a$ in a unital $C^*$-algebra $A$. If $\lambda_n \in \sigma(a_n)$ converges to $\lambda \in \mathbb{C}$, then $\lambda \in \sigma(a)$. Does the converse hold?
So if $\lambda \in \sigma(a)$, does there exist a sequence $\lambda_n \in \sigma(a_n)$ with $\lambda_n \to \lambda$?
I'm positive that it holds if $\dim(A) < \infty$, and if $A$ is commutative, but in the general case I don't see a proof. Perhaps some additional assumptions are necessary?

Converse fails in $B(H)$ (Problem 102 in Halmos's Hilbert Space Problem Book): take $a_n$ to be the bilateral weighted shift with weights $1$ everywhere except at the $0$-coordinate, which has weight $1/n$. The limit point $a$ is similar except that the weight at the $0$-coordinate is $0$. Then each $a_n$ has the unit circle for its spectrum, but $a$ is not invertible, so has $0$ in its spectrum.
