Monotonic version of Weierstrass approximation theorem

Question

Let $f\in\mathcal{C}^1([0,1])$ be an increasing function over $[0,1]$.

Prove or disprove the existence of a sequence of real polynomials $\{p_n(x)\}_{n\in\mathbb{N}}$ with the properties:

• $p_n(x)$ is a non-decreasing function over $[0,1]$;
• the degree of $p_n$ is $n$;
• $\|f-p_n\|_{\infty}=\max_{x\in[0,1]}|f(x)-p_n(x)|=O\left(\frac{1}{n}\right)$.