Certainly you can prove it if you make some simple assumptions. These are:
1. The laws of physics are the same in each inertial reference frame.
2. Maxwell's equations for electromagnetism are among the laws of physics
3. Momentum and energy are conserved.
There are then many proofs, each of which is I think a bit longer than we would normally include on this site. My favorite proof uses an old fact from Maxwell's equations, that for a beam of light E=pc, where E is the energy, p is the magnitude of the momentum, and c is the speed of light. Pick the right system emitting light rays and it turns out that you can use that to show E=mc2
, if by "m" you mean the inertial mass, the m that appears in p
, where v
I'm going to make that argument in class in a few weeks. It takes maybe 15 minutes to go through it very slowly with pictures.
(published on 02/04/12)