Give an example of a unital Banach algebra, $x,y \in \mathcal{A}$ and $r > 0$ such that $\|x-y\| < r$, while $\sigma_{\mathcal{A}}(y) \nsubseteq B_r(\sigma_{\mathcal{A}}(x))$.
Note: $$ B_r(\sigma_{\mathcal{A}}(x)) = \bigcup_{\mu\in\sigma_{\mathcal{A}}(x)}B_r(\mu) $$ where $B_r(\mu)$ denote the open ball in $\mathbb{C}$ with radius $r > 0$ and center $\mu \in \mathbb{C}$.
Here's my thought: I am aware that when $x$, $y$ are commuting then $\sigma_{\mathcal{A}}(y) \subseteq B_r(\sigma_{\mathcal{A}}(x))$. So I've tried to find a non-commutative Banach algebra, like $M_n(\mathbb{C})$. But then I'm stuck. How could I show that in this Banach algebra $\sigma_{\mathcal{A}}(y) \nsubseteq B_r(\sigma_{\mathcal{A}}(x))$? Any help would be appreciated!