= Solution
Write the <Bernstein polynomial> in its partition-normalized <Bernstein basis>:
$$
\boxed{B_n(f,x)=\sum_{k=0}^nf(k/n)\binom nkx^k(1-x)^{n-k}.}
$$
For $0\le x\le1$, every weight is nonnegative, and the <binomial theorem> gives their sum as $(x+1-x)^n=1$. Consequently
$$
|B_n(f,x)|\le\sum_{k=0}^n|f(k/n)|\binom nkx^k(1-x)^{n-k}\le\|f\|_\infty.
$$
Taking the <supremum norm> proves \b[$\|B_n(f)\|_\infty\le\|f\|_\infty$]. The construction yields degree at most $n$, possibly lower for special functions.
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