Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that the sequence of interpolating polynomials $p_n(x)$ of $f$ at those points satisfy $(-\infty,0]\cap p_n([a,b])\ne \phi$ for infinitely many $n$ ?

NOTE: If this is true, then it would imply that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there is a continuous function $f:[a,b]\to (0,\infty)$ such that the sequence of interpolating polynomials $p_n(x)$ of $f$ does not converge uniformly to $f$ on $[a,b]$. i.e. a positive answer to my question implies Faber's theorem.

anysequence of subsets $A_n$ of $[a,b]$ of size $n+1$? For instance can we exclude $\operatorname{diam}(A_n)\to 0$ as $n\to\infty$? $\endgroup$ – Pietro Majer Nov 1 '18 at 9:11