A Proof of the Central Limit Theorem Using the $2$-Wasserstein Metric

We prove the Lindeberg--Feller central limit theorem without using characteristic functions or Taylor expansions, but instead by measuring how far a distribution is from the standard normal distribution according to the $2$-Wasserstein metric. This falls under the category of renormalization group methods. The facts we need about the metric are explained and proved in detail. We illustrate the idea on a classical version of the central limit theorem before going into the main proof.