 Solution

Choose a sufficiently large positive $n$ for which $g_n=B_n^{-1}f>0$ throughout the interval. Expanding $B_n(g_n)=f$ givesTherefore the coefficientssupply the requested representation, in particular with nonnegative coefficients. This proves positive Bernstein coefficients for a strictly positive polynomial. Strict positivity cannot generally be weakened to mere nonnegativity: a nonzero polynomial such as $(x-1/2{)}^2$ vanishes at an interior point, whereas every basis term is positive there, so no nonnegative-coefficient representation could vanish unless every coefficient were zero.