Let f(x) be a continuous function defined for 0 ≤  x ≤ 1. Consider the functions (called B...

Let f(x) be a continuous function defined for 0 ≤  x ≤ 1. Consider the functions

 (called Bernstein polynomials) and prove that

and then use Theoretical Exercise 1.Since it can be shown that the convergence of Bn(x) to f(x) is uniform in x, the preceding reasoning provides a probabilistic proof of the famous Weierstrass theorem of analysis, which states that any continuous function on a closed interval can be approximated arbitrarily closely by a polynomial.

Theoretical Exercise1

Let Zn, n ≥ 1, be a sequence of random variables and c a constant such that for each  Show that for any bounded continuous function g,