Bernstein uniform approximation by polynomials : What is $E_n(f)$?

Question

I am having a difficult time understanding what Bernstein's constant is. Wikipedia states "Let $E_n(f)$ be the error of the best uniform approximation to a real function $f(x)$ in the interval $[-1, 1]$ by real polynomials of no more than degree $n$. In the case $f(x)=|x|$". What I am confused about is why $$\lim_{n\rightarrow\infty}2nE_{2n}(f)$$ is a constant and not a function. I thought that $E_n(f(x))=|f(x)-f_n(x)|$ where $f_n(x)$ is the best polynomial approximation. So, is there a more mathematical way to describe what $E_n(f)$ is or show some examples that makes it easier to understand what it is?