Uniform convergence of Bernstein polynomial derivatives ($\Vert B_n^{(r)}f-f^{(r)}{\Vert }_{\infty }\to 0$)

For fixed $r$ and $f\in C^r[0,1]$, uniform convergence of normalized finite differences to $f^{(r)}$, uniform continuity of $f^{(r)}$, and $(n{)}_r/n^r\to 1$ imply $g_{n,r}\to f^{(r)}$ uniformly. Positivity and partition of unity make Bernstein polynomials contractions in the supremum norm. Consequently $\Vert B_{n-r}{g}_{n,r}-f^{(r)}{\Vert }_{\infty }\le \Vert g_{n,r}-f^{(r)}{\Vert }_{\infty }+\Vert B_{n-r}{f}^{(r)}-f^{(r)}{\Vert }_{\infty }\to 0$. The differentiability assumption is essential; continuity alone does not define the target derivative.