# Monotonic version of Weierstrass approximation theorem

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 $n$;
• $\|f-p_n\|_{\infty}=\max_{x\in[0,1]}|f(x)-p_n(x)|=O\left(\frac{1}{n}\right)$.
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 It seem that the $O(\frac{1}{n})$ error is impossible, even if $p_n$ is not required to be non-decreasing. I did some simple google search, and found this. – Landscape Apr 20 at 18:27