Proposition 5 (Approximation by Bernstein polynomials) Let \(f: [0, 1] \rightarrow \mathbf{R} \) be a continuous function. Then, the Bernstein polynomials \[f_n(t) := \sum_{i=0}^{n} \binom{n}{i}t^i(1-t)^{n-i}f(\frac{i}{n})\]converges uniformly to \(f\) as \(n \rightarrow \infty\). This asserts that continuous functions on (say) the unit interval \([0, 1]\) can be approximated by polynomials.