  We give a new proof of the classical Central Limit Theorem, in the Mallows
($L^r$-Wasserstein) distance. Our proof is elementary in the sense that it does
not require complex analysis, but rather makes use of a simple subadditive
inequality related to this metric. The key is to analyse the case where
equality holds. We provide some results concerning rates of convergence. We
also consider convergence to stable distributions, and obtain a bound on the
rate of such convergence.
