Show that for every open set $\displaystyle U\subset\mathbb{C}$ that contains the spectrum $\displaystyle \sigma(x)$ of $\displaystyle x\in A$ where $\displaystyle A$ is a Banach algebra that there exists a $\displaystyle \delta>0$ such that $\displaystyle \sigma(y)\subset U$ whenever $\displaystyle y\in A$ satisfies $\displaystyle \|y-x\|<\delta$