= Solution
The <Bernstein polynomial> is the positive weighted average
$$
B_n(f,x)=\sum_{k=0}^n f(k/n)b_{n,k}(x),\qquad b_{n,k}(x)=\binom nkx^k(1-x)^{n-k}.
$$
For $0\le x\le1$, the <Bernstein basis> weights are nonnegative and the <binomial theorem> gives $\sum_{k=0}^nb_{n,k}(x)=1$. Hence $|B_n(f,x)|\le\sum_k\|f\|_\infty b_{n,k}(x)=\|f\|_\infty$. In particular,
$$
\boxed{\|B_n(f)\|_\infty\le\|f\|_\infty.}
$$
This is the contraction property of the <positive linear operator on continuous functions> defined by the <Bernstein polynomial>.
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