Limit of the spectrum in Banach algebra

Question

Let $A$ an unital complex Banach algebra, $a_{n} $ is a sequence such that $\lim_{n\to \infty}a_{n}=a$. What is the relation between $\lim_{n\to \infty}\sigma(a_{n})$ and $\sigma(a)$. I think that it could be $\sigma(a)\subset \lim_{n\to \infty}\sigma(a_{n}) $. But it is just a hope. I would appreciate someone help.