Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 11F / a / Solution

For $f\in C[0,1]$, define its Bernstein polynomialIf $X_{n,x}\sim \mathrm{Bin}(n,x)$, thenLet $M=\Vert f{\Vert }_{\infty }$. By uniform continuity, for any $\epsilon >0$ choose $\delta >0$ such that $|f(u)-f(v)|<\epsilon /2$ whenever $|u-v|<\delta$. Splitting the expectation according to that event and using Chebyshev's inequality gives, uniformly in $x$,For sufficiently large $n$ this is below $\epsilon$. Thus $B_{n}f\to f$ uniformly on $[0,1]$. An affine change of variable treats every compact interval, proving the Weierstrass approximation theorem.