For any function $f\in C[0,1]$, it is well known that there exist an unique polynomial $p^{*}\in P_n[0,1]$ such that $||p^{*}-f||_{\infty}\leq ||p-f||_{\infty}$ for any $p\in P_n[0,1]$. In this fashion, one can define an operator $A_n: C[0,1]\mapsto P_n[0,1]$ as $$ A_n(f):={\rm argmin}_{p\in P_n[0,1]}||f-p||_{\infty}. $$
For a fixed $f\in C[0,1]$, how to prove that there exists a constant $C$ depending on $f$ and $n$ such that $$ ||A_n(f)-A_n(\tilde{f})||_{\infty}\leq C||f-\tilde{f}||_{\infty},\quad \forall\tilde{f}\in C[0,1]. $$