Let $f(x)=\frac{1}{1+25x^2}$ and range is $[-1,1]$. Given $n+1$ equidistant points $x_0 = -1,x_1,...,x_n = 1$ and their values $f(x_0),f(x_1),..,f(x_n)$, perform polynomial interpolation by the $n+1$ points and get the polynomial $p_n(x)$. The question is
Show there exists some $x\in[-1,1]$ such that error $p_n(x)-f(x)$ does not converge to $0$ as $n\to\infty$.
I think it is possible that we need to show $\mathop {\lim }\limits_{x \to \infty } ({\max _{ - 1 \leqslant x \leqslant 1}}|f(x) - {p_n}(x)|) = \infty $, but I am not sure how to do this. I don't know how to express the error of newton's interpolation in a decent way. Thank you!