On the Rayleigh-Ritz variational method

We give a simple proof of the well known fact that the approximate eigenvalues provided by the Rayleigh-Ritz variational method are increasingly accurate upper bounds to the exact ones. To this end, we resort to the variational principle, mentioned in most textbooks on quantum mechanics and quantum chemistry, and to a simple set of projection operators. We think that present approach may be suitable for an advanced course on quantum mechanics or quantum chemistry.