Let us toss a biased coin that comes up heads with probability p and assume the validity of the Strong Law of Large Numbers as described in Exercise 15. Then, with probability 1,

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Sn as n - 0. If f(x) is a continuous function on the unit interval, then we also have )- fp). 12 Finally, we could hope that E (- (=). E)= f) - Show that, if all this is correct, as in fact it is, we would have proven that any continuous function on the unit interval is a limit of polynomial fune- tions. This is a sketch of a probabilistic proof of an important theorem in mathematics called the Weierstrass approximation theorem.