Suppose I have a $x_0<x_1<\dots<x_n$ and a Lagrange base polynomial $$L_i(x) = \prod_{i\neq j=0}^n \frac{x-x_j}{x_i - x_j}.$$
This answer states, that it holds $$ \bigl(L_i(x)\bigr)^2 \leq \frac K{K + (x-x_i)^2}$$ with a constant $K>0$.
How can I see that this holds? For which interval $I\ni x$ does it hold? How does $K$ depend on the $x_i$ and on $n$?