how to estimate maximum of Lebesgue function of arbitrary nodes?

Denote $$S_n=\left\{x_{n}=(x_{n0},x_{n1},...,x_{nn})|a\leq x_{n0} with $$-\infty and $$n\geq 1$$. For any $$x_n\in S_n$$, define the Lebesgue function $$L_{x_n}(x)$$ by $$L_{x_n}(x):=\sum\limits_{i=0}^{n}\left|\frac{\prod\limits_{0\leq j\leq n,~j\neq i}(x-x_{nj})}{\prod\limits_{0\leq j\leq n,~j\neq i}(x_{ni}-x_{nj})}\right|.$$ How to show that $$\inf\limits_{x_n\in S_n}\max\limits_{x\in[a,b]}L_{x_n}(x)\geq C\log n$$ for some constant $$C>0$$ ?