Approaching a continuous function with polynomials satisfying $\left|P_n\left(x\right)-f\left(x\right)\right|<\frac{2}{n}$

Question

For any continuous function $f\left(x\right)$ such that $\left|f\left(x\right)\right|\leqslant 1$,
and given $n\in\mathbb{N}^\ast$,
does there exist a polynomial $P_n\left(x\right)$ with degree $\leqslant n$ such that
$$\left|P_n\left(x\right)-f\left(x\right)\right|<\frac{2}{n}\quad\forall x\in\left[0,1\right]$$ If not, can any continuous function be approached by such polynomials at $O\left(1/n\right)$?

Answer 1

That's not true for a given fixed $n$. Because a continuous function can change very rapid and attain both $1$ and $-1$ in every sufficiently small enough interval.

When f is Lipschitz Continuous, $P_n(x)$ can be chosen to be the Bernstein Polynomials.