Uniform approximation of a continuous function on $[0,1]$ by polynomials with a control on the uniform norm of the polynomials

Question

Let $f \in C([0,1])$. By the Weierstrass approximation theorem, it is possible to uniformly approximate $f$ on $[0,1]$ by a sequence of polynomials $P_i$.

(i) Can we also require that $\|P_i\|_\infty \leq \|f\|_\infty$ for all $i \in \mathbb{N}$?

(ii) Can we also require $|P_i(x)| \leq |f(x)|$ for all $i \in \mathbb{N}$ and $x \in [0,1]$?

Note: this question is related but not quite the same.