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Let $P_n$ be a polynomial of degree $n$. For a function $f:[a,b]\to\mathbb{R}$, Let $\Delta(P_n) = \sup_{x\in[a,b]} |f(x)-P_n(x)|$. and $E_n(f) = \inf_{P_n} \Delta(P_n)$. A polynomial $P_n$ is the best approximation of degree $n$ of $f$ is $\Delta(P_n) = E_n(f)$.

If there exists a polynomial of best approximation of degree $n$, there also exists a polynomial of best approximation of degree $n+1$.

I Know this is old problem, But this problem Now I can't see any solution,can you help me,Thank you :

and there post this problem,but can't solution:There exist a degree $n+1$ polynomial of best approximation if there exist a degree $n$ polynomial of best approximation

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I also asked this problem but I think the claim is incorrect as the question and the previous one implies you can find a polynomial via induction which is not true. For example, the function $f(x) = x$ on$[0,1]$ has a minimal polynomial of degree 1 with $x$; if you added $\lambda x^2$ to it then $\Delta(\lambda x^2 + x) = |\lambda|$, which obviously has no minimum with $\lambda \neq 0$ –  user110503 Apr 30 at 8:34
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