7.3 Some ancillary results 6. Weierstrass's approximation theorem. Let f : [0, 1] - R be a continuous function, and let Sn be a random variable having the binomial distribution with parameters n and x. Using the formula E(Z) = E(ZIA) + E(ZIAC ) with Z = f(x) - f(n-'Sn) and A = {In 'Sn - x| > 8), show that n lim sup f ( x) - f ( k) " ) xk (1 - x)"-k = 0. n- 0