Maximizing the entropy of a sum of independent bounded random variables

Let X₁,...,X_n be n independent, symmetric random variables supported on the interval [-1,1] and let S_n=sigma_i=1 ⁿX_i be their sum. We show that the differential entropy of S_n is maximized when X₁,...,X_n-1 are Bernoulli taking on +1 or -1 with equal probability and X_n is uniformly distributed