  We prove that the uniform growth bound $\omega_0(\mathcal{U})$ of a discrete
evolution family $\mathcal{U}$ of bounded linear operators acting on a complex
Banach space $X$ satisfies the inequality
$$\omega_0(\mathcal{U})c_{\mathcal{U}}(\mathcal{X})\le -1;$$ here
$c_{\mathcal{U}}(\mathcal{X})$ is the operator norm of a convolution operator
which acts on a certain Banach space $\mathcal{X}$ of $X$-valued sequences.
