= Solution
For fixed $t$, let $B$ have the <binomial distribution> with parameters $m,t$. The <Bernstein polynomial> is $f_m(t)=\mathbb E[f(B/m)]$. Continuity on the compact interval gives <uniform continuity> and a finite bound $M=\|f\|_\infty$.
Given $\epsilon>0$, choose $\delta>0$ so $|f(s)-f(t)|<\epsilon$ whenever $|s-t|<\delta$. Since $\mathbb E[B/m]=t$ and $\operatorname{Var}(B/m)=t(1-t)/m\le1/(4m)$, <Chebyshev's inequality> gives
$$
|f_m(t)-f(t)|\le\epsilon+2M\Pr(|B/m-t|\ge\delta)
\le\epsilon+\frac{M}{2m\delta^2}.
$$
The bound is independent of $t$, including both endpoints. Taking the supremum and then letting $m$ tend to infinity proves \b[$f_m\to f$ uniformly on $[0,1]$]. This provides the required <polynomial>-approximation theorem with its proof.
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