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Let $\{f_i\}_{i\in I}$ be a family of smooth functions on the circle such that, for each $j\in\mathbb{N}$ there exists $C_j>0$ such that $$\left\lVert\frac{d^j f_i}{dx^j}\right\rVert_\infty<C_j$$ for all $i\in I$.

Question: Given any error $\epsilon>0$, can one find an integer $N_\epsilon$ such that for any $i$, there exists a polynomial $P_i$ of degree at most $N_\epsilon$ such that $||P_i-f_i||_\infty<\epsilon$?

Remark: It is clear that if the derivatives were not uniformly bounded in $i$ then this is not possible. It seems reasonable to expect that such an estimate is possible with that assumption in place, but I am looking for a rigorous proof.

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