[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let us toss a biased coin that comes up heads with probability [math]p[/math]

and assume the validity of the Strong Law of Large Numbers as described in Exercise Exercise. Then, with probability 1,

[[math]] \frac {S_n}n \to p [[/math]]

as [math]n \to \infty[/math]. If [math]f(x)[/math] is a continuous function on the unit interval, then we also have

[[math]] f\left( \frac {S_n}n \right) \to f(p)\ . [[/math]]

Finally, we could hope that

[[math]] E\left(f\left( \frac {S_n}n \right)\right) \to E(f(p)) = f(p)\ . [[/math]]

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.