# The smoothing property of Bernstein polynomial

The smoothing property of Bernstein Polynomial, proved by Kelisky and Rivilin in 1967, $$\lim_{k\rightarrow\infty}B\;^{(k)}\left(f;x\right)=\left(1-x\right)f\left(0\right)+xf\left(1\right)$$ can be seemingly borne out by the following result: $${\mathrm B}_{\mathrm n}\left(\mathrm x(1-\mathrm x);\mathrm x\right)=\left(1-\frac1{\mathrm n}\right)\mathrm x(1-\mathrm x)$$

Any concrete ideas?