  We prove an inequality relating the norm of a product of matrices $A_n\cdots
A_1$ with the spectral radii of subproducts $A_j\cdots A_i$ with $1\leq i\leq
j\leq n$. Among the consequences of this inequality, we obtain the classical
Berger-Wang formula as an immediate corollary, and give an easier proof of a
characterization of the upper Lyapunov exponent due to I. Morris. As main
ingredient for the proof of this result, we prove that for a large enough $n$,
the product $A_n\cdots A_1$ is zero under the hypothesis that $A_j\cdots A_i$
are nilpotent for all $1\leq i \leq j\leq n$.
