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For any degree-at-most-$n$ trigonometric polynomial $t_n$, linearity and reproduction giveThe operator norm bound in the preceding part yieldsTake the infimum over all such trigonometric polynomials. By the definition of best uniform approximation,No choice of a minimizer is needed for this argument. It is an instance of the polynomial reproduction error bound: a bounded linear reproducing operator has error at most $1+\Vert P\Vert$ times the optimal error.