  We give a direct alternative proof of an area law for the entanglement
entropy of the ground state of disordered oscillator systems---a result due to
Nachtergaele, Sims and Stolz. Instead of studying the logarithmic negativity,
we invoke the explicit formula for the entanglement entropy of Gaussian states
to derive the upper bound. We also contrast this area law in the disordered
case with divergent lower bounds on the entanglement entropy of the ground
state of one-dimensional ordered oscillator chains.
