A New Direct Proof of the Central Limit Theorem

We prove the Central Limit Theorem (CLT) from the definition of weak convergence using the Haar wavelet basis, calculus, and elementary probability. The use of the Haar basis pinpoints the role of $L^2([0,1])$ in the CLT as well as the assumption of finite variance. We estimate the rate of convergence and prove strong convergence away from the tails.