Erdős problem 1151

Let $\mathcal{L}^nf$ be the Lagrange interpolation polynomials of a continuous $f$ on the Chebyshev nodes. Prove that, for any closed $A\subseteq [-1,1]$, there exists a continuous function $f$ such that $A$ is the set of limit points of $\mathcal{L}^nf(x)$.