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**math8** Let $\displaystyle x_0,x_1,\cdots x_n$ be distinct points in the interval [a,b] and $\displaystyle f\in C^1([a,b])$.

We show that for any given $\displaystyle \epsilon > 0$ there exists a polynomial p such that $\displaystyle ||f-p||_\infty < \epsilon$ and $\displaystyle p(x_i)=f(x_i)$ for all $\displaystyle i=1,\cdots, n$.

I know $\displaystyle ||f||_\infty = \max_{x\in[a,b]}|f(x)|$ and I wonder if the polynomial they are asking for is the Lagrangian polynomial interpolating f at the nodes $\displaystyle x_1,\cdots,x_n$. If yes, I am not sure how to prove the problem.