Mathematics — Probability · weierstrass-approximation
Problem
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,
nSn→p
as n→∞. If f(x) is a continuous function on the unit interval, then we also have
f(nSn)→f(p).
Finally, we could hope that
E(f(nSn))→E(f(p))=f(p).
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 functions. This is a sketch of a probabilistic proof of an important theorem in mathematics called the Weierstrass approximation theorem.