  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.
