= Inverse Bernstein approximation on a fixed-degree polynomial space
{title2=$\|B_n^{-1}f-f\|_\infty\to0$}
On the fixed finite-dimensional space of polynomials of degree at most $d$, uniform convergence of $B_n$ on each monomial and <equivalence of norms in finite dimensions> imply $\varepsilon_n=\|B_n-I\|_{\rm op}\to0$. A <Neumann series> then gives $\|B_n^{-1}-I\|_{\rm op}\le\varepsilon_n/(1-\varepsilon_n)$ for sufficiently large $n$. Consequently the displayed convergence holds. A strictly positive polynomial has positive minimum on the compact interval, so its inverse Bernstein approximants are also strictly positive for all sufficiently large $n$.
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