Let $f$ be a continuous function such that $f(0)\ge 0$. Prove that there exists a sequence of polynomials $(P_n)_{n\in \mathbb N}$ with positive coefficients such that $P_n$ converges uniformly to $f$ on $[-1,0]$.
I tried using Bernstein polynomials but the coefficients aren't always positive, so if I can just approximate a polynomial with such a sequence then we are done because we can just use bernstein polynomials, but i am having difficulties finding a sequence that would work for $f(x)=-x$ for example.