# Earnshaw's theorem and 'free space'

I am trying to understand the proof for Earnshaw's theorem. Though the theorem states

> that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges ([Wikipedia][1])

In the proof, Gauss's Law in free space is being used (namely that the charge density $\rho$ is zero). How is that correct if we're looking at a collection of point charges? I feel I am being wrong on a very fundamental level.

[1]: http://en.wikipedia.org/wiki/Earnshaw%27s_theorem "Wikipedia"