Suppose that $A$ is a finite dimensional unital Banach algebra, $x\in A$ and $\{x_\alpha\}$ is a net (not necessarily sequence) in $A$ such that $\|x_\alpha-x\|\to 0$. Also we have a real number $K>0$ such that $$\|ax_\alpha-x_\alpha a\|\leq K\|a\|,\qquad (a\in A)$$ Could we conclude that $K\geq\|x\|$ ?