  In this note, we give a simple proof of the Borsuk-Ulam theorem for
$Z_p$-actions. We prove that, if $S^n$ and $S^m$ are equipped with free
$Z_p$-actions (p prime) and $f: S^n \to S^m$ is a $Z_p$-equivariant map, then
$n \leq m$.
