Earnshaw's theorem and 'free space'

[vetinari]

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"

 

[Goddar]

Hi. I think the argument is based on the fact that point-charges are represented by delta-functions, so that Gauss' law returns zero everywhere except at these sources where it actually diverges. Hence, there are no well defined minima or maxima of the field potential and no stable equilibrium...